How do I "record" live with reverb

Short answer - no, not record. Longer answer is that this is done for a reason, and that it can sort-of be got around - although you might not like the inevitable latency lag.

Audition will let you put an effect on an input and let you monitor it whilst you are recording, but the recording itself will be dry - this has always been regarded as good practice, since you can't remove it again afterwards - and it applies to all effects. And it can fool you. If you play back the track you recorded with an effect on it, the effect will still appear to be on the track - until you remove it. Then all of a sudden you realise what's happened!

If you have a multi-input sound device, then there's nothing to stop you taking a channel output with an effect on it as a direct output, and physically looping it back to another track and recording it there - that will print the reverb with your playing just fine, because it's not a track effect, but part of the input. And if you use that channel feed as a feed into Zoom, then you have at least got live playing with reverb - to do this, you just have to select a hardware output as a channel destination, rather than the Master fader.

But, you will get round-trip processing latency, and that's unavoidable. The best way to stop this being such an issue for you locally is to get as much of (all?) the dry return from the reverb out of it as possible, and just use the wet, where there's a delay anyway. And the trouble with this is that then you have to work out exactly what it is that you're sending to Zoom anyway...

I hope that this is of a little help at least - it's not an easy situation at all. Personally I wouldn't do it this way; I'd have a secondary feed from the mic on the trumpet fed to something like a small guitar amp in the room - via a hardware reverb unit, so you've artificially made the room a bit acoustically larger. But there again, I've got all this junk lying around so it's a bit easier for me. But if you could borrow the necessary, it would be a lot easier to set up than all this playing about with Audition and latency!