Julia gave great insights and articulate observations for kids this age. A great book is: "Your 3 Year Old" and you can get it at www.amazon.com It's a fast easy read and simply explains what each age-set is like so the Parent can understand THEM. I recommend it.
Okay you seem to have tried everything.
Bear in mind, that 3 years old is actually harder than 2 years old.
ALSO, gosh there is a baby in the house. He seems to regressed because of it? "Regression" in a child, means they are stressed. It's the way they cope, since at this age they don't have "proper" coping skills. Maybe he is just not adjusted to it all yet? Or he is not happy with the "expectations" upon him, as an "elder" sibling? Sometimes kids just feel "pressured" too by that, and they can't explain it.
Next, have you tried just talking WITH him and asking him what's up? For whatever is it worth, at his age and his ability to communicate or express his feelings... let him. Just listen, don't judge and see what he says. MANY times a child just has to get their yah'yah-s out... and they don't feel they can... or that anyone is really listening to them or that they are not "allowed" to express their feelings.
Next, teach him about "feelings" the names of it AND THAT EVEN ADULTS have bad days and grumpy moments... LET him express himself... BUT, emphasize that "We are a family... everyone has bad moods, but we TRY not to take it out on anyone..." Or, "We are a TEAM... can you try your best to HELP Mommy?" ie: this way, you are INCORPORATING him into the "solution."
Next, when he behaves this way.. .give no raised voices to it either, nor attention. If he flips out in public, say nothing, gather everything up and him included and go back to the car. Then sit there, he will probably be screaming... and repeat "Mommy loves you... but I don't like the screaming. We are going to sit here and relax now...can you HELP me?"
OR, you use a "toy time-out." ie: when he flips out, you put his favorite toy away... where he cannot get it. You CALMLY explain that his toy is on time out/or he can't play with it... until everything is calm. Then when he calms down, sit him in your lap, hug him, and talk with him saying "do you think your toy can come out now? Is everything feeling better? What do YOU think?" (this way you are 'teaching' him empathy and how to reflect on himself and problem solve and tell you his feelings...doing it together with you).
Next, is he just copying what other kids do? Does he go to preschool? Is he happy there? Does he have any problems? Ask him.... he sounds stressed and frustrated to me, for some reason. If he does not go to Preschool... PERHAPS he needs something just for him, and he may benefit by going part-time even, to Preschool? After all, his day is basically... Mommy with baby, Mommy nursing baby, baby crying, baby needing "more" of Mommy, baby baby baby... and him being maybe an after-thought. Although I know you said you make time for him... but maybe even that is not enough... and as A Mom with 2 kids myself... a child can feel "rushed" indirectly, even if you are just playing with them. ie: My daughter will SOMETIMES tell me "Mommy! I need attention...you are always in a hurry because brother... can't you just sit with me? I want to talk stories... just put him to nap already..."
Next, he seems to have trouble "transitioning" from one thing to the next. Which is common in kids. SO, for him... you might have to give him some head's-up and a "warning" that something is coming up AND what you expect of him. ie: "in 15 minutes, we are going to the store... can you help Mommy and get ready? Clean up? Put on your jacket? When we get there, please help me and stay close to Mommy and hold my hand when we are at the store..." This way, you are clearly telling him what is coming up, what to expect, and how you 'want' him to behave... for some kids, this helps.
And, if he cannot behave, then simply do not take him to the park for example.
He needs to learn boundaries like any kid. The thing that helps, is a ROUTINE. CONSISTENTLY. Some kids, need more structure... including for naps. Then over time, they LEARN what is expected, what is coming up, and it provides them with "cues" as to what they will be doing.
As for naps: put your baby to nap FIRST. Have a nap routine and a consistent nap time for baby, daily. (this will help you later in the long run). THEN, after baby takers her nap, be with him. Let him wind-down first to nap. Me, I make everything quiet, make the rooms dark, and it takes me about 1/2 hour to wind-down my kids for nap, before they actually go to nap. Kids NEED to wind-down first, AND need a verbal head's up about it. ie: In 20 minutes, we will get ready for nap. Put your jammies on." Don't give him a choice about it and say "Do you want to nap?" Just make a statement about it, letting him know what you expect. Then, IF he fights for naps...then just stop. For me, what I do EVERY SINGLE DAY... is, we have the same nap time and routines. And the same for bed. If my kids put up a protest (which they rarely do because I have been doing this routine since they were babies), is I stop. BUT, the rest of the afternoon is a QUIET time....and I keep everything quiet and I tell my kids that "It's Mommy's nap time.... so help me. No noisy activity..." or something like that and then I do NOT engage in adverse behavior. Period. They KNOW that about me. They keep to the "routine."
It's hard... but you NEED to do THE SAME THING EACH TIME. It is about repetition, routine, consistency, and for the child to learn and KNOW what is coming up, what is expected of them and then reinforcing it. For me, I do not use treats or rewards and do not like to "rely" on that. Only once in a blue moon will I do that. When I am desperate.
Another thing you can try is make a cute "chart." On that, make columns with the days of the week... and simply make a child friendly "schedule" for him. ie: Monday= wake up, eat breakfast, play time, help put away, read with Mommy, nap time, park time, music time, crafts time, etc. THEN, he will "know" what his day will be like and what is expected of him and what he will be doing. AND, when you look at it WITH him, you are 'teaching' him how to 'read' how to understand a 'schedule' and what not. Some kids like "structure" like this.. .and it' makes them feel special and knowing what it is for "them."
Well just some thoughts/ideas. I know it's not easy.
All the best,
Susan