Starting and idling problems

If you put the 6.1 injectors back in and it runs, then my first thought would be that the 6.1 injectors are bigger and giving it more gas and that's what it is needing.
The factory computer is going to give you a tune for a stock car, and that is assuming everything is correct as far as specs. You need to make sure that it is getting the proper fuel pressure, The sensors are giving proper signals. Coolant, Intake air temps are correct , map sensor etc.
For example, if your coolant temperature sensor signal is off 15 degrees telling the computer it is 70 when it actually 55 degrees, that may be just enough to lean it out and cause a starting problem.
Also these swaps are usually getting free flowing headers and exhaust systems that are going to change things.
When I talked to Hotwire they recommended a different tune over stock in these resto mod cars, he said they couldn't believe how much better they run and sound different with the updated tunes.