3yo son hitting/pushing at daycare - not sure how to get through to him

I'm at my wits' end daddit. My son has been getting into trouble for months over pushing/hitting at daycare. These incidents usually center around toy/territory possessiveness (e.g. pulling a kid off of a tricycle he decided was 'his', pushing someone trying to get into 'his' play house). It's getting to the point where he is being sent home in the middle of the day when an incident occurs. We just had a daughter last week, so this is getting to be very stressful, and I'm concerned if we can't nip this in the bud in the next month or two, he's going to get kicked out of the daycare (and we really can't afford a full-time nanny instead).

I'm feeling frustrated over the ineffectiveness of the parenting advice we've been trying to put into practice to address this so far. We don't ever use physical punishment at home, and we don't even raise our voice to him (at most, I'll use my 'stern voice' if I really need him to listen). If he hits us at home, we'll pull the hand away and emphasize that it's wrong to hurt people. We've been reading him stories about hitting - emphasizing taking deep breaths and calming down instead. We haven't used punishments (e.g. if you hit someone at school, then later you don't get <blah>) because we've read those are ineffective? (though please let me know if you've had experience otherwise)

And he seems to get it....when he's calm. But then if he's upset (at school over a toy, or at home sometimes if we're transitioning) he'll get physical. I feel like most of the discussions are trying to get at a level of maturity/understanding that will be helpful long-term, but right now I just need compliance - he just needs to stop. When I was growing up, hitting my parents was a line we just wouldn't cross, because we knew we'd get spanked. I don't want to use physical punishments, but is there another way to create that kind of 'third rail! - you just don't do this!' effect? Or does anyone have a particular approach that worked well for them in a similar situation?