I own about twenty Bluefish444 cards - Supernova, EPOCH, and Neutron. Some are the older 2k, some are the newer 4k. They all work perfectly with CASPAR, although I never have attempted a higher resolution than 1080p. I’m simply not sure if CASPAR supports any 4k resolutions on the BF444 cards, although I have used the cards @ 4k using the BF444 C++ toolkit.
The important thing with BF444 is to understand all the cards are flashable, so you’ve got to flash them with the correct firmware for your purposes. All three of those types of BF444 cards (Epoch, Supernova, Neutron) can be flashed as “capture” cards, so if your BF444 card is flashed for inputs only, then CASPAR will not output anything.
If you register your card with BF444, you will gain access to their tookit, and the toolkit contains a firmware flash program and all the available firmware. Re-flashing only takes a few minutes.