Oh, poor bub! Thank you for taking him in!
If your dog is screaming and wont stop, the situation is intolerable, of course.
If you are in an area where there is RVHD2, housing the new rabbit in the same room as the other rabbits would pose a risk. This disease is highly contagious. I dont see that you have much choice though, given the dog-situation.
There are several parasites that could spread between the rabbits, both parasites that live on the skin and internal parasites. Has he been seen by a vet? A vet can prescribe Revolution (selamectin), which is the best and safest anti-parasitic for rabbits. (And while were on the subject, a public service message that we always give: Never use any compound containing fipronil on a rabbit. Fipronil is sold OTC for dogs and cats, under a whole host of brand names (Frontline is a common one). It is deadly toxic to rabbits. End of public service message.)
Don’t let the other rabbits come into contact with new bunny directly nor with his pee or poop. If you can, set up some form of barrier between the cages to prevent particles from floating freely between the cages. A bed sheet, blanket etc if it’s not possible to make a hard barrier.
Wash your hands when you’ve handled new rabbit, before you do anything with your other rabbits, even refill food, hay or water.
This bun is very lucky to have been rescued. Thank you again 💗