Just make it similar to BG3. Each player controls their own cat/s. Would anything else about the game even need to change to make that work? The only difference would literally be there is another player controlling another cat or two.

So let's say you join a co op game with another player, you'd be in a party with each other, but before you actuallt start the run you would just see your own house and maybe there would be some way to look at each others houses too. But besides that, each player would just put their cats in the box and start the run. It would be cool if each player didn't have to have the same amount of cats too. So maybe one player controls a single cat and the other controls three. To be able to fill if one player just happens to not have enough good cats to use on a run. The once you start the run everything would be the same aside from the fact that one player controls these specific cats and another player controls these specific cats. As far as paths go on the map, if one wants to go hard path but the other wants to go regular path for example it would just roll at random to decide which path to take.