Well, I don't have a specific tool recommendation for this, but in my experience I've found it easiest to just implement this stuff myself.
- Easy enough to throw a feedback form into your app hidden somewhere and just link to it during the beta.
- For logins and such I just use a simple signup code system (with a simple admin UI for adding more signup codes)
- Bug tracking... depending on your audience they probably wont be entering bugs themselves, so use the feedback form for this. If your audience is developers then you can simply point them at your bugtracker or just have them email you and you can enter them -- 10-20 people isn't a lot of bug reports
- This approach is 'free' in the monetary sense :)
Just my approach, sometimes it's worth it to just bang out this stuff really rough in an evening rather than setting up/managing a separate tool which you'll only be using during the beta. When your beta is over, this way, you just delete the related code (or better, re-purpose it to facilitate free accounts for bloggers or people you meet at conferences and stuff).