With reference to a dose of aperient salts, the phrase like a dose of salts means very rapidly and thoroughly.

These are, in chronological order, the earliest occurrences of the phrase like a dose of salts that I have found:

1-: From Extract of a Letter from Calcutta, dated 30^th Aug. 1805, published in The Morning Chronicle (London, England) of Monday 24^th February 1806:
—Context: Richard Colley Wellesley (1760-1842) was the Governor-General of India from 1798 to 1805. At that time, the Governor-General of India was selected by the Court of Directors of the East India Company, to whom he was responsible. The East India Company had its headquarters on Leadenhall Street, in the City of London:

The expences attending Lord Wellesley’s first trip up the country, and his preparations for his second, amounted to lacs on lacs of rupees, that would be above all conception of an European’s estimation, he had so many boats, for months, waiting his pleasure to set off. It would appear to you incredible, were I to relate the sums this man has expended in his buildings, guards, &c. &c. […]—I think the accounts of expenditure during this campaign, will operate like a dose of salts on the Gentlemen in Leadenhall-street.

2-: From The Daily Chronicle (Philadelphia, Pennsylvania) of Saturday 30^th November 1833:

From the Boston Morning Post, of Nov. 29.
POLICE COURT.

Winner vs. Loser, for an Assault.—On Saturday afternoon, the parties, harboring felonious designs upon each other’s purses, engaged in a game of nine-pins.—Winner, according to his own story, which was not contradicted in court, in a short time touched Loser to the amount of $2. Loser then told him he was “bursted,” and requested the loan of some money, but Winner refused to afford him “facilities” for speculating on his own funds. Perhaps he was not very choice in the terms of his refusal, and Loser vowed he should accommodate him, or “take a licking.” He was as good as his word, and proceeded, with commendable deliberation, by turning a boy out of the alley, shutting the door, and slapping his cheeks with his open hand. The belligerents then rested upon their arms until Sunday, when they again met, exchanged a broadside of big words, and Winner again received a slapped face. This was too much for him, and, curling himself into the shape of an apothecary’s prescription, he swore, “if he slapped him again, he’d go through him like a dose of salts.” Loser stood aghast at the pestiferous threat and portentous attitude of his antagonist, and, eyeing him with a mingled look of fear and incredulity, mentally ruminated—
“What man dare, I dare:
Approach thou like the rugged Russian bear,
The armed rhinoceros, or the Hyrcan tiger,—
Take any shape but that—and my firm nerves
Shall never tremble.”
He was willing to incur the risk of an external application, but could not think of undergoing a purgation in this damp weather, and desisted from further attack. He pleaded guilty, paid a fine of $1 50, and the costs. The witnesses, personal friends of the parties, remitted their fees, and so the hatchet was buried.

3-: From a correspondence from Windsor, Vermont, published in Mackenzie’s Gazette (Rochester, New York) of Saturday 2^nd June 1838:

People were flocking into Windsor from the adjacent towns in Vermont, and also from many towns in New Hampshire. Young and old, all classes were warm advocates for Canadian freedom. Never was there so much unanimity on any question as on this. “Would Martin Van Buren ¹ say but the word” said one tall bundle of thews and sinews to me “I guess, we would walk through them ’ere Royalists in Canada just like a dose of salts,” and so, I believe, they would. No feeling prevailed during the last war on this frontier equal to what exists here.

¹ Martin Van Buren (Maarten van Buren – 1782-1862) was the 8^th President of the United States from 1837 to 1841.

4-: From a letter by Major Downing ², published in the New York Daily Express (New York City, New York) of Saturday 1^st December 1838:

He says, if I will only come on and stand by him we’ll go through all these troubles like a dose of salts.

² Major Jack Downing is a fictional character created by the U.S. humorist Seba Smith (1792-1868).

5-: Seba Smith used “like a dose of salts and rubarb [sic] in an over eaten stomach” in Major Downing’s First Speech, in reply to Mr. Bellowspipe in the “Downingville Convention”, published in the New York Daily Express (New York City, New York) of Saturday 4^th January 1840:

There had ben a Bank of U. S. a few years before, and its charter had run out and it was wound up, and in its place a lot of State Banks had sprung up, and pritty much every man had set up his own shinplaster shop. The people was gittin more and more disgusted, and jist looking round to cure the evil,—for sich evils have nat’ral cures, though one cure aint quite as Democratic as another. Some few folks who had money were jist going ahead over them who was blowing up, and supplying good paper money of their own making, and by fixing agencies at different pints, jist as they do all through Europe, taking care as they do there, to keep their own iron chests to hold the gold taken in exchange for their credit,—and never trusting any one on their industry and good morals or as readily as they do men who have good gold property for security—and if the Democratic Party of that day hadn’t seen the drift of things—or if they acted then as the party calling itself Democratic now act—what would have been the condition of things? Why simply this. We should have a few old Rothchilds—Hottingeurs and Hopes—all everlasting rich Bankers—grown fat by the use of their Credit—besides a good many who have been blown up—but they holding the gold whilst the People hold the Bills—But the Democratic notion of that day didn’t think this exactly the thing for a republican people—they knew the people couldn’t get on without using credit, and that they would use it; that bad credit would break down all the enterprise of those who were starting into life with no other capital but their industry, and that such would, in a few years, become a mass of loafers, just as you see in all hard money countries; whilst those who were born to fortunes would alone be trusted by the Bankers—and so the rich would become richer, and the poor poorer; the old fashioned Democrats saw this, and they put a stop to it right off, by creating a Bank of U. States, and the Government took one-fifth interest in it, and I do truly believe, if the Government then had not ben so poor in money matters itself, it would have taken a bigger dip in it; but it took one-fifth, and made a putty good bargain in paying for it, and so managed the charter as to keep an eye on the business of the Bank, and having five Government Directors always in the Bank, so that nothing could go wrong, and on any complaint, walk the Bank up right before Congress, and if any thing was wrong hatchel e’m right off and correct it.
This Bank went right to work, clearing out the shin plaster folks, and there was sad work for a spell among e’m; it was like a dose of salts and rubarb in an over eaten stomach; but it all work’d out right, and they kept things smooth and regular as long as its charter lasted.

The phrase has come to be also used in the extended form like a dose of salts through + noun. The earliest occurrences of this extended form that I have found are:

1-: From the Dodgeville Chronicle (Dodgeville, Wisconsin) of Thursday 30^th March 1865:

From the Twenty-First. A private letter from the 21^st Regiment, dated Fayetteville, N. C., March 12^th, says Lieut. Morgan and Captain Ewen, two officers of the regiment who were taken prisoners at the battle of Chickamauga, escaped from the rebels near Columbia and came into Sherman’s lines. The regiment had a pretty hard march, but was in good health and condition. The letter graphically describes the march of Sherman’s army and its effects, by saying, “We have gone through South Carolina like a dose of salts through a sick child. Fifty years cannot wipe away the marks of the ‘vandal invaders.’”

2-: From the Wood County Reporter (Grand Rapids, Wisconsin) of Thursday 27^th November 1873:

Runaway.—Dr. Witter’s Bucephalus an untamed fiery steed that would have made Mazeppa’s mouth water, ranaway [sic] one day last week. Pills flew as thick as shot in a field engagement, and Johnny, who grooms the beast, went with the pills and things, like a dose of salts through a skeleton. We believe that friend Porter, the obliging station agent of the W. V. R. R. was also aboard the vehicle and was deposited beneath a dray, experienceing [sic] however, no injuries.

3-: From the Bismarck Semi-Weekly Tribune (Bismarck, North Dakota) of Wednesday 11^th April 1877:

Don’t be too Smart.

It isn’t a good plan to be too smart. At least two persons disposed to boast have learned better since they came in contact with Bismarck sports. One of them exposed to view a large roll of bills on every convenient occasion, told how he had made three and four hundred dollars a day gambling and said he would just like to see the man who could get away with him. Well, perhaps it would not be best to say the boys organized a little ring and went through him like a dose of salts through a sick horse, but he was effectually cleaned out and left town the next day a sadder if not a wiser man. It isn’t a good thing for smart people from the East to boast of their knowledge before they learn the ways of the West, for the chances are that those who make gaming a profession can show them tricks worth a half a dozen of theirs. No man coming to these western towns who minds his own business and carries a level head will ever have cause to complain of the treatment he receives, but the man who boasts of his smartness is very liable to get his eye teeth cut.