Tiamat is a great villain with one major flaw, and that flaw is named Bahamut. What kind of threat is the queen of evil dragons if she is opposed by an equally powerful (or more powerful) king of good dragons?
I haven’t played the 5e Dragon Queen adventure path: I hope it answers the question “if Tiamat gets free, why can’t Bahamut clean up the mess?” I hope the answer is not “regrettably he is busy on an unspecified engagement” or “he could totally do it but he doesn’t meddle in mortal affairs” or “let’s hope no one brings it up.”
The way I see it, Bahamut needs to be defanged, declawed, and de-breath-weaponed to allow Tiamat to shine as a campaign villain. Here are some ways to do it.
Bahamut is dead
Either just the other day or in the legendary past, Tiamat murdered Bahamut. This one is fun for PCs who worship Bahamut because they get the angsty nobility of championing a lost cause. Bahamut worshipers keep all their powers – he is no less divine for being dead – but his faction is powerless against Tiamat’s hordes.
Bahamut is crippled
As above, but Tiamat, unable to kill Bahamut, cut off his wings or blinded him in some ritually irreparable way. This one is nice because Bahamut can still give aid to the PCs and can still match Tiamat in a melee, but isn’t mobile enough to force a battle. Bahamut’s best move might be to take the form of a human paladin (a blind or lame one – maybe he’s Sir Isteval, that guy in all of WOTC’s Sundering products) and rally mortal support.
Bahamut is imprisoned
As I understand it, Tiamat is at half-strength and looking for a way to break her shackles and assume her full power. Maybe the same is true of Bahamut. Maybe the siblings imprison each other, like worms of ouroboros biting each other’s tails. If The PCs fail and Tiamat gets free, the world still has one slim chance: the PCs must brave Tiamat’s draconic dungeon and un-shackle Bahamut.