Are there cat trainers like there are for dogs? My instincts say get another cat! But it sounds like your partner may not have much patience left for it. If your cat is that angry all the time, he may just be happier in another living situation. It would be an act of love to let him go to another loving home. Not an act of dispare you should feel guilty about.
Vet here! Cats are tricky and can be quite the nuisance when stressed. How many litter boxes do you have? You need one box per cat plus one so he should have 2. Unfortunately neutering him late could be part of the problem. Definitely DO NOT get another cat like suggested above, that will just make the behavior worse. And no, there are not trainers for cats but there are things you can try. First, is there anything, other than the kids, that have changed around the house? Construction, new animals, new litter products, etc. You can try giving him some more stimulation. A window attached catio is a great option (just make sure you get him on heartworm prevention). Next we talk supplements and medications. Purina Calming Care is a food additive, I have seen some success but it does take 6 weeks to build up in the body. Solliquin is also good and natural. Feliway spray and plug ins are magical. Try all 3 of these together even but start with the Feliway. If after 6 weeks the supplement
Route is not working, try asking your vet for gabapentin and/or fluoxetine.
Damn. That's a rough one. I think you are right in rehoming, maybe just the way you go about it might make it better. Instead of just sending the cat to any old shelter, seek out sanctuaries or go through friends of friends connections so you can attempt to know he has a good home. It might take a bit to fully accept this decision and that's not something you can rush, no matter how quickly you want an answer. Do you have anyone close that can take the cat in for a bit? If it makes you feel better, we had to give away our bird. Wrecked my husband, I took on the leg work... Maybe have someone else do it for you? Good luck ❤️