I think I found a way to get it working after a few hours of trial and error. I downloaded Dxtory and was able to record only my PTT audio, nothing in the background, which is what I wanted. I used alt for it's PTT, which is my direct chat in TS and in game. All I did was edit my hotkeys for TFR to include alt in the modifiers. I have a G700s which I use the thumb buttons on to use my radio. Before I had O set to long range and U set to short range (because finding spare hotkeys in Arma isn't easy haha. Never use my watch anyway), with shift as a modifier to use additional channels. Now they're just set to alt+O and alt+U. Since people in your vicinity can hear you when you're talking on the radio regardless of physically using direct chat, I don't mind using direct and the radio simultaneously. I was also able to get both game sound and my headset sound working. The only minor gripe I have is that anytime someone joins, leaves, gets kicked, whatever, that sound gets picked up through TS going through my headset. Other than that, I'm pretty happy with what I have and I can deal with that one tiny issue. Thank you all for the help and especially to the LT for that youtube suggestion. That's what allowed me to get the software all set up.
If I'm playing a different game where all this hotkey stuff isn't needed, Shadowplay will be my go-to.