That way, when the recording hits the public airways
I can't imagine that this "recording" is not broadcast to the opposition in real time. Perhaps they just post it publicly after the fact?
If it's not done in real time and unless the person is dumb, they aren't going to upload any audio that asks them by name how they are doing. They will do one of a few things:
1) splice just that part out - easy give away, unless someone other than spy is recording the spy's broadcast and then posting the audio and doesn't splice that part out, in which case it's an easy give away and they rooted out their own spy
2) randomly splice random parts of the audio (including where you ask them by name how they are doing)
3) not share the audio at all - hard to do this one if it's being shared in real time and someone else, other than the spy, is recording it and posting it publicly.
Since they most likely will not post any audio with their name in it asking them how they are doing, you could use this to your advantage, because you could space out the same question to everyone on your team and since only you would know exactly how the question was spaced out between each person on the team and in what order, only you would know when there would be audio splices missing depending on who the culprit is.
You don't need anything fancy to do this. Maybe just a text doc and a repeating alarm on a wrist watch (every 2min alarm beep). Just do some key word/phrase that's a key word only to you in terms of timing to start things off. "Alright guys, let's rock this boat".
"Rock Boat <key word/phrase" @ 8:55PM //Start repeating alarm
//Watch beeps, ask a person on TS how they are doing
Asked <name> how they were doing @ 8:57PM
Asked <name> how they were doing @ 8:59PM
Asked <name> how they were doing @ 9:01PM
At the end of it, you're going to know in relation to the key word/phrase exactly what time it was whether or not anything recorded in TS is time stamped or not or 3rd party recording program or not. You're going to notice a spliced out audio x mins into the audio from the key word position, maybe even a few random spliced out audio segments as their attempt to throw you off, but all you have to do is record the whole segment beginning to end and you'd be able to match up from your whole recording to the spliced recording and figure out the times where the audio is missing.
You could probably do other tricks too:
Run TS on a separate computer running a really low resolution and use fraps to record the whole the screen including the little clock in the bottom right corner. That'd allow you to have all the audio and a time piece to gauge from. I'm not familiar with TS so I don't know if TS allows for time stamp logs of when who said what.
You could start recording early or make an attempt to have your team on TS 10mins early, and gauge by when people joined TS as to who possibly the spy could be. Use some kind of music track and the fraps idea above to help figure out when the spy first joined and started broadcasting your conversation. Odds are they are dumb and record beginning to end from the time they join, or pretty close to it.
Another possibility is doing a private message to each person of nothing more than a number 1 through 100 all tracked or logged in a text document. Private Message someone the number 72! and then a slight pause and then say "whoops!" as if you Private Messaged the wrong person, and then don't say anything else to them. On your side, you'll have 72 associated with a very specific name and you can just listen for your # + Whoops on the recording.