Hi again. I just thought I'd update this here (why not, I guess). The problem was with DOSBox. A person updated DOSBox for PSP, I used it, and the game started saving finally (
https://www.reddit.com/r/PSP/comments/crfeuf). Thank you again those who tried to help!
(The game is also running or at least starting up much faster with a mod I made for the DOS version of Fallout to include most Windows patches on it (F1DP), so I think it's finally playable (still slow, but at least doesn't take infinity booting up). Not too interested in playing it there anymore though. Maybe some other time)
Cheers! May the thread die in peace now ahah.
Edw590