"I think Dewi Lingo has a headache."
Translation:Dw i'n meddwl bod pen tost gyda Dewi Lingo.
So is bod a form of the verb to be or does it simply introduce a subordinate clause, like "that"?
Bod is what might be called the "infinitive" form of "To be" in welsh. It changes very irregularly and as you said can be used to mean that.