You have gotten some really great advice! My kids are older but my dd1 was very much like this as a 3-4 year old. it was extremely challenging. And like you I was juggling her and a baby. I think a lot of what she did was to get a reaction out of me. I was so, so tired. If she was naughty it would make me immediately look at her and comment on what she had just done. I resorted to changing the baby in my locked bedroom for awhile with her pounding on the door. I told her sorry, but you are not helping me with the diaper changes and you are making it harder for me. If you want to help or watch quietly I will let you in next time. Didn't always work though-- she would push the rules as far as she could and always find that exception to the rule.
It is also a pain to follow through with the rules sometimes. When dd was at that awful age, she would go to the playground and be really rough with the younger babies. Once, she pushed a newly walking baby over. She also used to pull binkies out of their mouths. This was embarrassing for me and I would go over and tell her stop and talk to her, which is probably want she wanted -- attention and also she could not help herself with what she was doing-- was too excited. So before we went the next time I told her if I saw touches that were not gentle, we would go home. Well we go there and she made a beeline towards a toddling baby and pushed him on his chest so that he sat on his bottom. He was not hurt but I was mortified. I picked her up and told her we were going home. I said sorry to the mom of the baby, too. It was inconvenient for me- I had taken awhile to pack our things, had a baby in tow that I had to deal with along with a tantruming 4 year old, and I had to get both of them out of the playground and into the carseats and then deal with more screaming at home. After a day or two we went back and she was better. Not perfect, but never pushed a baby down again.
Some ideas for you:
1) Your son sounds extremely intelligent and he is a fast thinker. I wonder if he could benefit from one-on-one time where you can sit down with him and teach him things. My daughter was very much this way and really benefitted from me starting to homeschool her at a very early age-- teaching her to read, write her letters, and we would read books that made her think. We also did a lot of printable art projects. This site was fantastic http://www.dltk-kids.com/ and also starfall.com for teaching to read. It kept her busy but required a lot of work from me. You can also involve him in helping with cooking tasks. Just keep him busy and on task sometimes. Yes your cooking will take 2x as long but he will feel he's getting attention and he will be able to fill some of those sensory needs. My dd's favorite task was shredding lettuce leaves. She also liked to set the table.
2) You absolutely need to have "me" time! Even if once a week when dh gets home, go out to the coffee house or bookstore or library, or see a movie with a friend.
3) Take your son on mommy and me dates. I think a lot of his attention getting is because he feels he is competing with the baby. Take him "big boy" places without the baby. Do the same for your oldest daughter to make it even. My kids love mommy dates and it was very special for them to have my undivided attention. We used to do simple stuff like ice cream or donuts or my youngest used to like to just window shop at the mall for an hour.
4) Set up a reward system for him to earn a special treat. Have a jar and put a penny in it every time you catch him doing something helpful. Make a big deal about it so he gets that attention. When he gets a certain number of pennies he can have a special treat. Don't take pennies away for bad behavior. Just say "I feel sad that you were not able to get a penny right now because you were not being helpful when I was trying to change baby's diaper." I work as a Peds nurse. I love that sometimes they set up a reward system for kids at the hospital who need to cooperate with something. They get three stickers for doing the desired tasks. When they get the third sticker they get to pick a present out of the surprise box. The box is in their room the whole time so they can see it, and they have cheap toys gift-wrapped to make it really exciting for them to get a prize. Maybe for his age a sticker chart would work better than pennies.
Hang in there! I am happy to say that my daughter-- who I was really worried about what kind of person she would grow up to be-- has turned out to be a very wonderful young lady of 12 years old, and is doing everything she should be doing at her age. :-) But man it was tough trying to work with her as a toddler/preschooler.