I have been using TB and avast! since several years with different mail providers and till now have never faced any issue. They act very nicely together and avast! is able to take care of whatever mail settings I setup in TB.
In fact TB will not allow loading of any remote content, so together with avast!, I feel quite safe. Also in the event that 1 e-mail does have a problem, avast! can delete that specific e-mail without deleting the whole e-mail folder which is what AVG seems to do.