Thanks for all of the help guys. I rode the trainer again this evening and it seemed to decrease the thumping by moving the roller a touch further away from the wheel. I did have a look this AM at the wheel while spinning it, and I couldn't see any noticeable change over the course of each revolution in terms of distance from the brake.
I will try remounting the tire tomorrow. Since no one has said that it seems like a big issue, I am less concerned. It isn't really all that irritating, and I can deal with it as long as it's not something that could cause long term damage or something (based off of the responses I would guess not). So I'll probably take off the tire and examine it to satisfy personal curiousity, if anything.
Thanks again! I will report back if I notice anything irregular or getting worse...