OK, I found away around the issue. It seems (at least for the half-dozen or so games I tried) that as far as moving around saves go, games only care about the PSN account the game was saved under. I had originally intended to "launder" my saves through PSN cloud storage, but it looks like that won't be necessary.
First thing, I ran a backup on my PS3. Then under my main ID that I already attached to my PSN account, I re-synced trophies and backed up my saves to USB (just to be safe). Then I deleted that ID and went to one of my sub-IDs. I attached that ID to my PSN account, synced trophies, and ran the games so I'd have saves created under the PSN account. Then I backed up the saves to USB, deleted the ID, and went on to the next one.
After I went through all of my IDs, I restored the backup I had created and copied the saves I had backed up from my sub-IDs to my main ID. Afterwards I was able to load the game saves without any problem.
I don't know if it was necessary to go through all the trouble, or if it would have been possible to load non-PSN account saves under a PSN account ID, but it seemed to work for me.