Why did the conspirators chose Oswald to be their patsy?
Because he was a self-described Marxist who had defected to the USSR and had recently been to the Soviet and Cuban consulates in Mexico City?
If so, when did they start patsy-ing him?
Oswald was the perfect patsy.
A marine who defected to Russia living in Dallas in the '60's?
A full blown Commie working in the building the shots were fired from?
There could not have been a more perfect patsy.
The moment he was arrested in the TT it seems like everyone in the DPD knew he'd killed Tippit and the President.
There was something about Oswald that was extremely 'hateable'.
When did they start "patsy-ing" him?
It might be that he was just the right man in the right place at the right time.
That said, I've always been struck by this picture:
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It shows Oswald handing out leaflets in New Orleans.
In the background is a man who looks identical to Bill Shelley - same body-type, same dress sense, same hair and same face.
This pic is a still taken from a short piece of film footage. The footage gives the distinct impression that Oswald is part of a larger group of people who are all stood around talking business while Oswald does the donkey work.
In the scenario I'm proposing ex-CIA man (if you can be "ex" with the CIA) Bill Shelley is the main man on the ground in Dealey Plaza. It is a very uncomplicated plot - fire a rifle at the passing limo. It requires a patsy who has been fooled into handing over his rifle and a shooter (who could make a good Patsy #2 if Patsy#1 doesn't work out).