While I don't know what state you live in, unless you have both parties consent to record a conversation it would not be allowable as part of your case.
The problem is you are only hunting for ways to make your EX look bad. You are so busy doing that you have STOPPED doing things that make you look "good".
If you are only awarded joint custody (and keep this up and that will be iffy at best) the ONLY person you have to blame is YOU because the ONLY person you can control is YOU.
And I really have to wonder how badly you want full custody. IMO if that was your main desire you would have stopped this terrible and damaging behavior a long time ago and really focused on the goal at hand. Instead you made it all about your EX, her off the books pay, her foul mouth, her dirty father and his GF, the OM, the fact you chose not to move and get a better paying job and the fact your EX is not a good shopper. All that BS for what? Not only has it gotten you nowhere but it now may get you much less.
I hope one day you are honest with yourself and really examine why you chose to keep acting this way when custody of your CHILD was at stake. If your true and pure goal was to obtain full custody ALL this other stuff would have been nothing more than tiny blips on the radar that meant nothing to you.