I am midway through a migration from the internal database and its been a real ball-ache. Considering that this is a MUST DO process for every single customer, it shouldn't be like this. In client $$$, its a £750 job. That's not right.
I suggest radically improving this experience or having TeamCity pick a more robust database from the beginning. On Windows, it should be SQL Server Express, which is redistributable, on Linux, I guess you can go for MySql.