Black bears I know a little about. I have a hunting partner that tried to whack one with a 45. I cannot begin to tell you what his load was. I can tell you he now carries a 44 Ruger. He did kill the bear, with my 308. An agitated bear with hounds etc is harder to kill than a relaxed bear. I would guess a bear looking to eat you would also be somewhat agitated.
I did have a she bear with cubs charge me, twice. She came at me, ran away, then came back again, then went up a tree with her babies. I didn't feel much fear at that time. I do remember reaching for the bolt snap on the hounds, thinking she ain't gonna get us all tied together.
Had a bear charge me once, turned him with a 308 under the chin, through everything that he needed to function and stopped under his hide by the hip. Knocked him down, but he got up. He was under 200 pounds dressed. A bear is a tough animal.
IMO a 45 is far superior to a sharp stick.
I have no experience with the Appalachian trail. Are there that many bad guys out there? Few will call a 45 under gunned for a BG.
I know about bears that have been trained to take food packs from hikers in the Porcupine Mountains. They are destroyed by the Rangers. Hikers train the bear. Goes like this see a bear, drop food pack and split. Yogi likes picinic baskets. Some become very insistent about the packs. It makes it easier for the Rangers to figure out which one to neutralize.
!0 years ago or so a black bear tried to make off with a boy scout in NW Wisconsin. The lad had a brave leader that ran the bear off. A houndman was called out the next day and he neutralized the bear.
I would say you have a better chance of being hit by lightning than have a bear try to eat you. However, should a bear be trying to eat you, a 45 would be better than nothing. It won't do you any good in your pack, that's for sure.