Either they sent you a bad ballast (always possible), it was really cold when you installed the ballast (since they seem to have issues in the cold), or the ballast isn't really the problem at all...I'm going with the third choice.They sent me a replacement ballast, the bad one was discarded... There is still an issue.
TRS even sells the relay harness so I don't understand why they don't recommend using it.
Now do our cars have the "Bulb out" warning light? I don't remember seeing it. Maybe they don't recommend the relay harness if you are using the Canbus harness, but that shouldn't matter either way.
I hope you get it resolved, I know it's frustrating when stuff like this happens.