I think I managed to pinpoint the root of the problem with Fallout saving in PSP: the
minimum requirements for Fallout under Ms-Dos dictate that it needs
32MB available RAM, double of which it requires under Windows - 16MB there. Have a look
here.
As a matter of fact, PSP does also have exactly 32MB RAM, for example you can look
here - using its memory for some of the PSP's system workings already takes away some amount of it. Seeing as DosBox itself also needs some amount of memory for operating, not sure how much exactly, you can see how that puts Fallout at a disadvantage - the setting
memsize=40 could not even go above the actually available memory after all the factors, which would be somewhat lower than 40MB. And that might explain the troubles with saving game more than once. Fallout 1 simply runs out of memory to operate correctly.
I suspect that playing other games in DosBox on PSP, which (usually for Ms-Dos games) require less memory, shouldn't be a problem unlike with Fallout 1.
P.S. having said that, since I don't know how PSP handles the writable storage (the specs page mentiones just the disc drive which is playback only), and what size it is, that might be also a cause of the problem, lacking enough space to handle saving games.