The Romans won the social war otl because they basically just started promising citizenship to anyone who surrendered. Which is what the Italians wanted in the first place, so they should start doing that here...
Galatia should already be owned by Mithridates, and more as a personal fief than a vassal state. About 10 years ago (I think) from the beginning of the TL, he gathered all the Galatian chiefs for a feast and had them killed. The Galatians, with no leadership, had no choice but to submit.
Tigranes has his own boogeyman (the Parthians) to worry about. Without first taking the Parthians out, Tigranes has no choice but to remain east of the Euphrates; with no threat to the east and plenty of menace on the west, Mithridates will maintain peace with his ally and son-in-law Tigranes.
At least, that's if nobody goes crazy/nonsensical.
Tigranes and Mithradates were kind of a dynamic duo. Tigranes was Mithradates' son in law, and OTL they were extremely close, backing each other in wars, invasions, and Tigranes even included Mithradates' symbol of a falling star on his coinage.