Prove REGULAR_TM is undecidable
Look up Rice's teorem. Its proof is quite short and sweet. It says that any non-trivial property of a TM language (i.e., one that not all TM languages share) is undecidable, in the sense that it can't be decided by looking at the TM if the language it accepts has the property.

The property of being regular (or a context free language, or the empty language, or the language of all strings, or...) is non-trivial, so if a particular TM accepts a language with the property is undecidable, by Rice's theorem.