I successfully killed the level 52 Handsome Dragon! It took a while, and I did die once, and I'm glad that's over!
I'm answering my own question because comments are too restrictive in length and formatting for my response.
Thanks Keith for your suggestions, the key takeaway I got from you was to keep constant pressure on the dragon, and because of that, I realized that the dragon wasn't actually flying out of the area, just very high so I was losing visibility of it from where I was taking cover at the end of the bridge near the entry gate. Once I began tracking it to keep some sort of damage going, I saw what it was doing and was able to use the SMG and assault rifle to keep firing while it was soaring to the top of the building - where I assume it would regenerate health if I hadn't kept the damage going.
Some other things I learned:
On my playthrough, which is the first time through the DLC, I
couldn't simply run across the bridge to enter Dragon's Keep, I had
to defeat the dragon to continue the quest.
There are two ammo shrines and several ammo crates near the center
of the bridge, which I hadn't initially noticed while maintaining
cover at the entry gate end of the bridge. When the dragon loops
under the bridge or soars to the top of the building, take the
opportunity to rush out and restock. A good shield and
health-regeneration class mod help in case it bombards you with fire
while you're out of cover.
After you confront the dragon, you'll initially be on your own.
Eventually Tiny Tina will summon Roland to help out in the fight.
In my case, Roland never threw down a turret, but you can watch him
to help determine the dragon's position.
The dragon has a pattern of attack where it sometimes hovers over
the bridge. A launcher comes in handy then, as well as near the end
of the fight to help knock down those last health points, slag it
and switch weapons when you see an opportunity to use the launcher.