Originally and chiefly Australian English, the phrase a bastard on Father’s Day is used to express unluckiness or unhappiness.

The earliest occurrence that I have found is from The Four-legged Lottery (London: T. Werner Laurie, 1958), a novel by the Australian author Frank Hardy (1917-1994)—as quoted by Gerald Alfred Wilkes (1927-2020) in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990):

bastard, happy (lucky) as a ∼  on Father’s Day Unhappy, unlucky
1958 Frank Hardy The Four-legged Lottery 128: ‘I’ve got about as much luck as a bastard on Father’s Day.’

The second-earliest occurrence that I have found is from The Boston Globe (Boston, Massachusetts) of Sunday 5^th July 1970—the Australian tennis player Ken Rosewall (born 1934) had been defeated at Wimbledon by the Australian tennis player John Newcombe (born 1944):

Rosewall, after losing to Rod Laver once in Boston, expressed the way he felt yesterday: “I felt like a bastard on Father’s Day,” he said that day.

The sense of the phrase is unclear in the following extract from The System, a short story by the above-quoted Australian author Frank Hardy, published in The Sun-Herald (Sydney, New South Wales) of Sunday 23^rd January 1983:

Cameron proposed a toast to his infallible roulette system, then he scoffed down his soup like a noisy Australian bushman, poured more champagne, ate caviar on toast with white onions, served the ladies graciously their pepper steaks and frites, tossed the salad and generally behaved like a bastard on Fathers’ Day.

In Six of one and half-a-dozen of the other, published in The Sydney Morning Herald (Sydney, New South Wales) of Tuesday 11^th September 1984, Alan Peterson noted that there are two opposite interpretations of the phrase:

What do you understand by these expressions:
Cop is sweet;
Better half;
Happy as a bastard on Father’s Day?
The staff of the Macquarie Dictionary has found that these phrases, although made up of simple words, have different meanings for different people.
[…]
In a cottage in the grounds of Macquarie University the dictionary staff has been tabulating replies to questions asked in the newsletter of the Macquarie Dictionary Society. […]
[…]
As for the phrase happy as a bastard on Father’s Day, the newsletter asked: Does it mean (a) very happy, or (b) very unhappy? Very unhappy, said 12. Very happy, said four people, apparently reasoning that no father meant no yearly present.
Professor G. A. Wilkes, in A Dictionary of Australian Colloquialisms, says the term means unhappy or unlucky. One of his examples is from Frank Hardy’s Billy Borker Yarns Again (1967): “The Parrott would be happy being talked out of seven winners in succession?” “Happy as a bastard on Father’s Day.”

In the review of The Loser Will be Later to Win (Carlton, Victoria: Pascoe Publishing, 1985), a collection of stories, sketches, reminiscences, by the above-quoted Australian author Frank Hardy, John Hanrahan criticised the repeated use of a bastard on Father’s Day and of other phrases—review published in The Age (Melbourne, Victoria) of Saturday 18^th January 1986:

Hardy dines out on the vernacular and that is one of his strengths. But he often serves it up cold and old. He is not a writer of whom one often says, “I wish I had said that”; rather I feel often that “I’m glad I didn’t say that”.
[…] In consecutive stories, a character is so thirsty that he could drink through “an Afghan camel-driver’s cock rag”. In three different stories, a character behaves like “a bastard on Father’s Day”. So Hardy moves from cliche to expressions that stand up and dance at you. Even extravagant dances lose their appeal with insistent repetition.

I have found a British-English use of the phrase, in an extended form, in the account that Andrew O’Hagan wrote of a fortnight in Peru with a group of British tourists, organised by Explore Worldwide—account published in The Guardian (London and Manchester, England) of Saturday 23^rd January 1999:

Peruvians are partial to a bit of flattened guinea-pig. It comes roasted on a plate, its teeth on edge. Bob from Sunderland * gave it the once-over. He wasn’t up to it. “I feel like a prostitute’s bastard on Father’s Day,” he said, reaching for a slug of local the local cerveza.

(* Sunderland is a city in the north-east of England.)