homebrew can be placed on FAT32 USB, but I always recommend using an SD card for homebrew (apps/ folder on SD) and data on usb (iso, roms, etc.)
it'll prevent issues you might have if you are using only usb and no SD.
some homebrew have SD card hardcoded too, so it'll not work with usb at all. (I'm not sure about syscheck, maybe it works with USB? in doubt, just use an SD card).
you can use a small SD card, no need of 256GB!
just go with 16 or 32GB SDHC card, instead of SDXC which might not work.
you really don't need a lot, even 2GB or 4GB is enough, but not sure you'll find any.