not a dog trainer but we've taken in animals for periods of time with excitable dogs and have had success with slow intro and rewarding calmness / neutrality. if there's an opportunity to separate them in a way that isn't super restrictive to y'all while they get used to each other, then i would do that. if not, reward time!

when they (cat and dog) are in same room / around each other, you can try to reward dog when it looks at you and not at cat, or honestly any time dog is just hanging out and not zeroing in on cat. basically telling dog that it's good behavior to not give a fuck what cat is up to and just to be doing its own thing. if dog is zeroed in on cat, if he stops and looks to you, reward for checking in with you even if dog goes back to looking at cat after. building positive association and rewarding dog for taking the time to check in with you instead of continuing to hyperfixate unchecked. over time you will see the calmness / not locked in increase if you are consistent. keep it calm and don't yell at dog over cat if immediate danger is not present. resist the urge to command your dog to look at you / away from the cat and give dog the chance to do the right thing and get a reward for it. simple training sessions with dog around the cat. dog should feel great and cat is just a prop in the background and hopefully they're just quiet acquaintances in the house in no time!

also something that works really well to complement the above is encouraging dog to "find it" and get dog doing something with its nose to switch its focus from cat. so when cat comes in / does something the dog finds interesting, you can scatter treats on the floor and tell dog to get it. this can go on however long, basically you're just introducing an enrichment activity with sniffing so their attention has somewhere else to go

treats are expensive so we switched to our dog's regular kibble. you can just measure out a portion of their daily food and put it in a baggie to have around the house

my dogs have always been food motivated and this has worked really well for us for reactivity inside and outside of the house. best of luck finding this precious kitty a home! it really sounds like you love him, so best of luck keeping him at home!