I've just recently tested out this Funtuna fork on a MechaPwn-unsupported FAT PS2 (SCPH-39003), with a single Chinese, non-MagicGate 2×128MB (256MB, split into 2 pages) in both memory card slots, and it works without issues.
However:
Funtuna, the PS2 Memory Card browser exploit that corrupts the Memory Card icon-loading process, doesn't work if you have a Sony PSone memory card inserted into a PS2, as that exploit also relies on a file date that is stupidly far into the future for it to work:
2099/12/31 23:59:59
Problem is that Sony PSone savedata DOESN'T have date/timestamps, breaking that -Tuna exploit in the process, as the Sony PlayStation 2 treats such data as the sort that supercedes the aforementioned date, to the best of my interpretation of this phenomenon.
This means that I would suggest that if anybody wants to use that exploit, ONLY insert a -Tuna-exploited PS2 memory card at boot, until the exploit has been loaded & DON'T insert PSone memory cards or any memory card with a date/time surpassing the -Tuna exploit.
If, however, you ignore this advice, you will not be able to make the exploit work, and you will need to reboot the PS2 WITHOUT ANY MEMORY CARDS THAT BREAK THE EXPLOIT INSERTED INTO THE CONSOLE PRIOR TO EXPLOITING IT, but fortunately, making any mistakes mentioned in this thread will not result in the files on the -Tuna-exploited memory card, or any others, getting corrupted in the process, so there is no need to worry.
That said, another phenomenon I have noticed here is that if you have BOTH a Free McBooted memory card AND a Funtuna'd (with Free McBoot) memory card inserted into the PS2, and you initiate the Funtuna exploit, instead of the Funtuna exploit being initiated to start up Free McBoot, it instead launches (u/w)LaunchELF. This only happens when the Free McBooted memory card is in SLOT 1 and the Funtuna'd memory card in SLOT 2, BUT NOT VICE VERSA, and fortunately, this will not corrupt or cause harm to any data on ANY of your memory cards.
On that note, as far as my knowledge goes, the boot priority of the PS2 is as follows, based on trial and error:
CDVD drive > Memory Card Slot 1 > Memory Card Slot 2 > HDD/SSD in Expansion Bay > USB drive (if anything does get booted up from there, but I doubt it - feel free to correct me if I am wrong).
UPDATE: I've found two additional things since writing this particular post - and a little quirk
1. inserting a PSone/PS2/DVD/CD WILL ALSO BREAK the -Tuna exploit's initiation process, requiring you to take out the offending media (just like taking out the PSone memory card or any PS2 memory card with filesaves date-stamped newer than the -Tuna exploit), exiting the Browser and going into the main menu (black screen with orbs), and then re-entering into the Browser with ONLY the -Tuna memory card.
2. If you have any Japanese-language saves that are saved onto a -Tuna memory card, attempting to access/highlight them while in the Browser WILL CRASH YOUR PS2. Just reboot it and you'll be fine, BUT make sure such saves aren't on the same memory card as the -Tuna exploit-installed one if you don't want any problems.
3. Accessing a -Tuna memory card that is installed with use for SCPH-50000+ consoles on a pre-SCPH-50000 FAT or certain older SCPH-50000 consoles (my testing involved 2 SCPH-50000s, one with a DVD player version 3.00, which crashes, and one with a DVD player version 3.02, which works okay) WILL CRASH YOUR PS2. Again, reboot it, and you'll be fine.