I don't see why it follows that, if Knox was guilty, he should have been responsible for the first two frame ups. In the bonkers world of Ellery Queen anything goes. How about this...?
Imagine Sloane sees Knox carting the body about. He doesn't know who it is, but he's afraid that investigations will show he's got the best motive. He decides to frame Khalkis, if only to get the case closed as quickly as possible.
Knox knows that someone has framed Khalkis but doesn't know who. He needs the case to stay open long enough for him to mention the existence of the painting (the culmination of his plan being the fake theft) so he explodes the frame by revealing that he was the third man that night.
Sloane, informed that Knox was present that night concludes that it was him that he saw moving the body, now blackmails Knox.
Knox, to remove the complication of blackmail, now arranges to frame Sloane and kill him but, thinking that there's at least one other framer at large who may come after him (he doesn't know that it was Sloane) he decides to exonerate himself once and for all with the clue of the 1000 dollar bill.
As for how it got in the watch; he originally planted it there at the time of the murder as a clue by which he could frame someone else later (pick someone, anyone, by this point it doesn't really matter who) but when the clue goes undiscovered he appropriates it for his own use. We only have his word for it that it's his bank note. He phones his bank manager himself and writes down the serial numbers for Ellery, but he could be phoning anyone or writing down any numbers. Ellery checks the serial numbers against the list but never bothers to confirm with the bank (despite being descirbed at the point when he checks the serial number as "a thorough young man").
Then he embarks upon his fake blackmail plan to steal the Leonardo from himself.
True, it's nuts, but I don't think it's any more nuts than anything else in the book.
Hahaha, this is hilarious! But, although I might have some objections, this doesn't seem completely impossible.
But it gave me an idea: Try to outdo Ellery!
Take any EQ-novel (or short story) and give an alternative solution that fits all the given facts at least as good as Ellery's solution does.
It needs only fit everything up to the "challenge to the reader" (or equivalent), for otherwise it would be impossible in most cases, since Ellery's solution is often confirmed afterwards in the books.
The "challenge to the reader" is irritatingly pompous, and the claim that there is only one logically possible solution is certainly false if we interprete it literally, and it is almost certainly falso also if "psychological observations" are allowed. Therefore, Dannay and Lee would need to be taken down to Earth (yeah, yeah, they are dead.. oh... no pun intended ).
So, can you outdo Ellery in some of the novels?
As for myself, I have an idea for The finishing stroke. Unfortunately, I don't have the book available right now, but when I read it. I got the following idea:
John III is the killer. He killed John I, and convinced the world that he was John I by bribing the haidresser/barber, who was a poor man with a large family, to say that he had a mark on his head, normally hidden by hair, that John I was said to have...
Not sure it would hold, though, and I don't have the book available right now to check it more closely...
But maybe you can do better with this or some other EQ-novel?