well you know what the thing is.. when you're not doing so hot in training, companies will usually give you a chance to resign rather than fail you outright. all the airlines know this, so I think if they heard that an applicant resigned during 121 training, they will at least suspect that you were "asked to resign" aka failed out (even if you technically didnt.) even if you resigned on your own accord without being asked to. what I'm saying is, if you resigned because you were asked to, your records won't say "asked to resign." it will say, "resigned." and it says the same thing "resigned" if you resigned without being asked to. there is no distinction between the two on paper, so it's easiest to assume you were asked to resign, which is practically same thing as failing out.
basically you would have to do a good job of explaining it in person and convince them why they should pay to put you through training again.