I am treating it as an indoor tree.
As much as I cut back, it keeps sending its branches straight up, and it
doesn't back-bud. The main trunk also does not thicken appreciably.

Yup, that's what pomegranates do. Your only hope is to plant it in an oversized
nursery pot & see if that will do anything for the trunk. To get more
branching, about all you can do is cut back the top & use a branch as a new
leader. I tried thread grafting, but that didn't work either. You need to keep
trimming off those vertical shoots, except in the summer when you want it to
bloom.
Pomegranate is a subtropical & does better with a winter rest. If you can't put
it outdoors, next winter strip off the leaves & put it in the refrigerator for
a month or two.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)