After getting an update from borgqueenx, the DOSBox app is back, and this time it's getting a patch from original author CrazyC. Though there was no changelog in a strict sense, he did say that bug fixes have been applied in this version.
Here's the short text that the developer posted in his release:
I've fixed most of the bugs that I could find in the 3.x build of DOSBox (at least with the fat psp). Now you should be able to exit and set any about of memory in the conf file as long as it can be allocated with malloc. Also, with a pair of prx's it has the same functionality as the 1.5 build. Neither prx is necessary to run the program.
Download: DOSBox PSP 06/25/0 patch