There is not a place where you can push on that makes it exquisitely worse. The type
of TN that would be confused with TMJ is the Type 2
or the "Symptomatic TN of MS." When I had TN the trigger tooth would hurt, but so would the whole jaw and the pain went all the up to the TMJ - which makes sense, because that is where the Trigeminal nerve emerges to give sensation to the face.
Now, there are three branches of the Trigeminal nerve (Cranial nerve V) These are V1, V2, and V3.