Simply by deleting the content of the .../Cryptic/Star Trek Online/Playtest/ folder. I wiped out everything except the screenshots. Then next time you log in and select Tribble it downloads a new copy of the client. But as I said, it only worked on the first login for me. After you've been on Tribble and logged out, expect to have to reinstall it again the next time you want to go on there. That's not a complete disaster when you're talking about a test server, but we're about one or two weeks away from that problem migrating to Holodeck, and going through an hour long process of reinstalling and patching they game every day seems like a bad thing...
Anyway, my first guess was that the on demand patching bit was causing the trouble - afterall the game works on the first run, which seems like something it wouldn't do if it were a hardware compatibility thing. It works, it runs, and then breaks itself in the process. But disabling on demand patching didn't help. Still, I haven't tried disabling it on a clean install yet. So I guess I'll try that next, and I'd suggest you try the same. Fingers crossed.
The crash does occur in safemode, yes. With no more or less in the way of explanation than oherwise. And no, I haven't got anything called Teamviewer installed. From the way you phrased it, that would've effected Holodeck and Tribble both. I have (thus far) had no trouble logging into Holodeck. It's only Tribble. Which depressingly enough is Holodeck's future.
Anyway, I tried what I suggested to Temerson - reinstall and patch with on-demand patching disabled - and again it worked fine on the first login. I logged out without even having gone past character selection, and the immediate login attempt after that resulted in another mystery crash. So that made no difference at all.