I dont think theres a question on any application that says "have you ever been asked to resign." The question is worded "have you ever been terminated." I say you can definitly play this situation out to be totally harmless to your career.

Because you resigned after IOE, your PRIA records will be clean (you should confirm this to make sure the timing of your resignation doesnt result in it showing as a training failure). Secondly, when they ask why you resigned, you can cite personal reasons (death in the family) as an honest an accurate answer.

dont tell anyone that you performed poorly in training, especially since theres no way they can find that info out unless you tell them! Now go get em tiger! And yes, I should have become a scam artist instead of a pilot