Belmonts have to be involved in the overall story IMO. They don't have to be an invincible family with inexplicaply one male heir to inherit the whip and save the day, but, to quote a past CV game have "a long history of battling evil". I would rewrite them as experts of fighting vampires and such, but not the only group to do so. They don't have to be the center of the story, but a significant element.
There has to be a "Dracula", too. Since the Stoker/real life Drac/pop culture icon and his Dr. Wily Syndrome have eventually caused problems, I would not necessarily call him Dracula and I would seriously rewrite him. But I would keep his best attributes: a tragic character. A flawed antihero turned villain. An intelligent and sophisticated, yet cruel man. He would be a schemer. Intimidating and ruthless and boiling hatred under his regal appearance and manners. Not someone who lets things get to point all he can do is lose to someone not even a match for his powers.
He would not wait in his castle to be cornered. He would make bitter enemies out of those who challenge him, using psychology and brutal means against them. He would succeed in killing Belmonts and other major characters. I'm not sure he should have a blood feud with the Belmonts again. And he would come very close to wiping out the humans he hates.
He would not begin another die-reborn-cycle. He might die a couple of times, but eventually he would either somehow change or be out of the picture completely. He would be a major villain, but not the main one. In fact, my version would not have any single central big baddie.
I would also keep at least the big six of Belmonts; Simon, Trevor, Leon, Julius, Richter and Christopher.