As it turns out, the problem is invalid - there can be Turing machines with the given specifications that can accept a non-regular language. Consider the following thought experiment:
- Consider a word consisting of a's and b's, in which no b comes before any a. Let there be an additional character c in the beginning of the string.
- Now imagine a Turing machine that iterates through the c, then the a's, and replaces the first b with a character that makes the machine go to the right in the accepting state, r for instance. The machine goes to the left after writing this character.
- It now comes upon the last a, which it replaces with a character that makes it go the the left, l for instance. The machine goes to the right after writing this character.
- The machine then goes back and forth, replacing r's with l's and l's with r's, depending on which direction it is going.
- It will halt and accept once it reaches the initial c. (Which it will have replaced with a corresponding symbol in the very beginning of the process)
In this way, the Turing machine will accept the language of words, which contains a subword consisting of a c, followed by n a's, followed by at least n+1 b's. This is obviously not regular, because DEA's can't count. (Or more formally, use the pumping lemma to increase the number of a's to be greater than the number of b's)
Sorry for all of the confusion - this was actually a homework question! (And I assumed that it would be valid)