One of the Thompson boys should use his middle name or initial. That is how it is normally handled to distinguish two celebrities with the same name. However, the lesser celebrity is expected to acquiesce and use his/her middle name. With egos involved with these two boys, I got a feeling neither is willing to blink first!
IIRC, the Screen Actors Guild actually required different names for every acting member of their union. I have heard that is why Michael J. Fox uses the "J" - though apparently the original Michael Fox never enjoyed much success.
I don't know if that rule applies now that SAG has merged with AFTRA. I recall all the "Mark Elliots," "Chris Elliots," and "Rick Shaws" in various radio markets.