Maybe a firmware upgrade to your CSMKIII would resolve the issue? That is, assuming you are currently not running the latest one.
It could also be that the card itself does not make proper contact with the motherboard's cpu fast-slot.
Finally, go to the CSMKIII early boot menu and see if there might be a setting there that you could potentially change/tweak, in hope to resolve this issue (like something in regards to the uw-scsi bus, or memory related, etc.).
ahh...success
I cleaned the CPU socket with CRC Electronic Cleaner and now it boots sweetly

The socket must have gone dirty or something while this A4000 was stored for 3 years in a cupboard at my old appartment.
I did blow the sockets clean with canned compressed air before my troubles started to be sure that no dust had accumulated in the sockets
I upgraded the firmware on both my CS PPC and CS MKIII back some years ago when I was really fanatic about Amiga hardware and could have signed up as chief for the club of "who can collect the most Amiga hardware in 6 months" hehe