You should send him over to talk to my now-5 year old. She was like that at 3. Every day. Maybe she could give him a few pointers!
Seriously, though, I think that in a lot of ways it's a 3 thing. DD1 did leave a lot of that behind once she was past 4. I'm sorry you're going through this. It sounds like the worst part of all is your disagreement with your DH about how to handle it.
I agree with you-- I think spanking, or any other out-of-control response like yelling or getting involved in grappling with a hysterical child, are all terrible ideas. They only escalate the situation. A child who's having big, huge, out-of-control feelings needs to be handled with calmness, and steady consistency. My favorite approach, when DD1 started getting extremely oppositional with me, was always just to walk out. I'd say, okay, that's fine, you call me when you're ready. Then I'd go downstairs and go do something else. She's rage for awhile, sometimes a long, long while,
and then realize nobody was listening, and eventually she'd come find me, and be in a much better frame of mind.
In the case of not staying out of the sibling's room, what about a gate? They make really tall virtually unclimbable ones. We used a gate for DD1 when my twins were younger, so that poor DD1 could get some peace and privacy.
It helps, too, to really choose your battles. Relying on a more playful approach when you can, and distraction, and saving the "firm" approach for those few situations where you absolutely must set a limit. Letting the small details go, and giving him as many chances as you can to make his own choices and do for himself.
In the meantime, though, I think that you and your DH have some issues to work out. I myself would be looking for outside counseling, if my spouse was hitting my kid and I was against that hitting. That could so easily turn into scary abuse, if your DH loses his temper-- and an oppositional 3 year old can CERTAINLY push those buttons.