As far as I know, there is no hard rule governing the styling of the trademark symbol aside from having it be recognizable.
However your colleague is probably "correct" in that setting the trademark symbol in the same typeface and styling as the Trademark itself visually implies that the symbol is part of the mark. It now falls upon someone to decide if the Trademark includes the symbol or not. If it your company, then this is important.
This is virtually the same argument people have about the inclusion of punctuation within a quotation which is not actually part of the quotation but rather part of the normal grammar and style of the language. Unless one has a very specific use e.g. string literals in computer science, one is probably going to have to give weight to the majority opinion.
Another similar issue is where a work of art is a portrait painting and the title line contains a date. If the date is italicized, it is part of the title, but it might not be the date of execution. If it is not italicized it is the date of execution of the painting and cannot be part of the title.
- Bob Smith, 1973 (ambiguous)
- Bob Smith, 1973 (title)
- Bob Smith, 1973 (not title)