Does this work with any sort of FAT32-formatted USB drive?
A frugal install of Puppy is basically copying vmlinuz (the kernel) and initrd.gz (the initial ram drive, basically a little Linux that gets copied to ram) and the sfs files (the Puppy files compressed in a squash file system.
That really is all there is to it. For Fossapup, that would be 5 files. They are just files that can be copied, moved, deleted, zipped, like any other ordinary files. Whether you copy the files yourself or a Puppy installer program copies the file doesn't make much difference, as long as the files get copied.
You need to start Puppy with a boot loader, but you already have a boot loader installed to start BionicPup.
So you do not need to install it again. You can just add a FossaPup selection to the boot loader configuration file.
My usb flash drive is an old one, with maybe 8 or 10 Pups on it. I suspect the old syslinux boot loader on it does not understand gpt partition tables, but it doesn't matter, It boots from the usb drive and finds vmlinuz and initrd.gz on the usb drive. The Linux in the initrd.gz does understand gpt, so it can find the sfs files.
My usb flash drive is formatted FAT32.
I have Secureboot disabled and Legacyboot enabled.
To install Puppy you just need to copy/paste 4 or 5 files.
If you already have a boot loader installed, you can just copy/paste a few lines in the configuration text file to add a new choice when the boot loader executes.