The verb handbag means, especially of a woman, to bully or coerce by subjecting to a forthright verbal assault or criticism.
—Cf. also handbags at ten paces.

Originally used with reference to Margaret Hilda Thatcher (1925-2013), British Conservative stateswoman, Prime Minister from 1979 to 1990, the verb handbag is first recorded in The Economist (London) of 7^th August 1982:

One of her less reverent backbenchers said of Mrs Thatcher recently that “she can’t look at a British institution without hitting it with her handbag¹”. Treasury figures published last week show how good she has proved at handbagging the civil service. And not just the poor bloody infantry in the clerical and executive grades. The mandarins have caught it as well.
Mrs Thatcher is already the most successful bureaucracy-cutter of all postwar prime ministers. Since coming to power in May, 1979, she has slimmed the civil and diplomatic services from 732,300 to 659,300, a reduction of 73,000 or 10%.

¹ The more common American-English equivalent of the noun handbag is purse.

Tyler Marshall mentioned the verb handbag in Thatcher: Crusader With Subtlety of Buzz Saw, published in the Los Angeles Times (Los Angeles, California) of Tuesday 26^th May 1987:

Her purse has become a symbol of aggression somewhat equivalent in potency to the country’s nuclear deterrent, and it has given the language a new form of assault, “handbagging.”

Its phonetic proximity to the verb sandbag, also meaning to bully or coerce, probably reinforced this signification of the verb handbag; this was mentioned in a letter published in The Observer (London) on Sunday 12^th November 1989:

Walleted by the handbag woman
Noting, near John Silverlight’s article on Nouns as Verbs (last week) a letter referring to ‘handbagging’, my wife, in the interests of anti-sexism, proposed the headline ‘Lawson wallets Thatcher’ as a counterbalance to ‘Thatcher handbags Lawson’.
But no, the attempt fails — the feeble thud of a wallet is as nothing to the crunch of a well-swung handbag, with its satisfying echoes of ‘sandbag’. What meaning then, in the open-ended game of verbing and nouning (even more entertaining than constructing the widest-possible split infinitive) which helps us to pass the time in the Ivory Towers?
Handbag, sandbag, wallet, wally . . . yes, clearly, ‘to wallet’ is a characteristic activity of Thatcherite society. An example is at once to hand in the adjoining article by Mary Warnock on universities’ funding. Obviously, the universities have been well and truly walleted by the Government. Who’s next?
Tom Brissenden,
Swansea.

illustration for Walleted by the handbag woman
The Observer (London) – Sunday 12^th November 1989

 

This use of the verb handbag stems from its literal meaning, which is, of a woman, to batter or assault with a handbag. The earliest instance of this literal sense that I have found is from The Arizona Republic (Phoenix, Arizona) of Thursday 8^th May 1952:

Music Critic Handbagged
Paris, May 7 (AP)—A woman in the audience—apparently a friend of the composer—handbagged a man who clapped before the end of the playing of Pierre Boulez²’ piece for two pianos Wednesday night at the Comedie des Champs-Elysees.
It was the first incident of the current program “Masterpieces of the 20^th Century” going on here.
Boulez, a 27-year-old Frenchman, was playing his piece, “Structures,” with a senior composer, Olivier Messiaen³. From the first notes the audience began to chuckle, but when one rude man began applauding boisterously before the end, the woman admirer let him have it across the face with her handbag.

² Pierre Boulez (1925-2016), French composer and conductor
³ Olivier Messiaen (1908-92), French composer