I don't think what you want is similar to the Halting problem since the grammar of regular expression. Considering the alphabet and the language recognized by your automaton is finite, you can still use a dummy algorithm that would try every world of your language and test if the regular expression is able to recognize it or not.
In practice, this method has an awful complexity but you don't have any "undefined" state you would have in Halting problem since the number of inputs is enumerable.
I actually don't know if a better version of this dummy algorithm exists, but i hope i answered about your question on similarity to the Halting problem.