It would certainly have been difficult for Denethor,
PrinceOfTheHalflings.

While in the short term he would have had some leeway, as Aragorn would not press his claim to the kingship until the War was over, there would be a clash over whether to send a force to the Black Gate to distract Sauron from Frodo and Sam. I've a feeling that Denethor would tell Gandalf that he wasn't going to risk Gondorian lives to support his bright idea of sending a witless halfling (sorry, two witless halflings) into Mordor.
Assuming that the Ring is destroyed, without needing the help of the force sent in the original time line, then the end of the War would mean that Aragorn makes a formal claim to the kingship. Faramir would, as you said, be on Aragorn's side. (I'm assuming that he and Éowyn still meet up in the same way.

) Éomer would also be a supporter, though I'm sure that Denethor would tell him to mind his own business, and not interfere in an 'internal Gondorian matter'.
Denethor would insist that all the legalities be followed, with a meeting of the Council of Gondor where Aragorn can properly make his claim. Things would be extremely difficult for Denethor if he is unable to persuade the Council to reject the claim. I think that most of the Council, including Prince Imrahil, would be in favour of Aragorn, not just because of the legitimacy of his candidacy, but because he had demonstrated his ability to defeat the enemies of Gondor, like Eärnil II.
While I agree completely,
Prince, that '
He [Denethor] clearly didn't want to be Aragorn's Steward and I'm not sure that Aragorn would have wanted him either', I don't believe that Denethor would commit suicide for that reason. In
LotR, he committed suicide because his mind had become unhinged after seeing the huge forces ranged against Gondor, and the seemingly mortal wound of his only surviving son. In this situation, Faramir would be alive and well, and Gondor victorious.
I agree with
Legate of Amon Lanc here, that Denethor would probably resign the stewardship in favour of Faramir, then retire into private life, refusing all honours. He would die '
somewhat forgotten', but would certainly survive as a controversial figure in Gondor's history, perhaps referred to as 'Denethor the Unwilling'. As you said,
Legate,
I can already see Aragorn at his funeral and Gandalf giving speech to the Hobbits about him on the way home, something like "he was a great man, but his time has passed and he was bitterly aware of that, yet his wisdom (etc...) after all it was his strength which held Mordor at bay for for half a century..." or something like that. Just try to imagine it, if I try, it's almost appearing in front of my eyes!
I can also forsee some rows in the future, with Elboron defending his grandfather's behaviour to his parents and to others.