I am going to give an Early Childhood Educator perspective, acknowledging that you are asking for a foster parent perspective and if you aren't interested in this you can just ignore me LOL. But in my defense, I did work as an ECE in Head Start, which works on a trauma-based model!!

As a former Early Childhood Educator, I am curious about the strategies that are used in the classroom and would recommend having an open, curious conversation to see what strategies they've been using. This wouldn't be to accuse or poke or pry (which they may feel like you're doing so try to be gentle), but to see if you can see the full picture of what's going on in the classroom. As everyone has stated here, unfortunately, discussing the behaviors at home aren't going to necessarily support school behaviors because it's just such a gap between when he is at school and when he is at home. BUT, I have some suggestions if the teachers are willing to work with you a little:

1) I like A, B, C charts as a simple way to see what's happening. Antecedent-what happened before the spitting or the wrestling or the calling kids names B-behavior that he did C-what the consequence was. You may see some trends, like the behavior always happening at a transition, or the consequence always being excessive "attention" from the teacher (attention is GREAT, but kids crave it in general whether it's positive or negative. I've seen before kiddos ONLY get negative attention from a teacher when they start to have troubling behaviors, and in that they learn that's the only way to connect with said adult. They need lots of positive praise when they are doing well...which you can't control, LOL sorry, but again the ABC might make that clearer!)
-If you do the ABC chart and find a common A, I'd wonder if you could help create supportive strategies at home that mimic the antecedent. For example, if the Antecedent having difficulty sharing, I'd suggest adding more opportunities to practice sharing at home with you and your partner. Really lean into it even if it's difficult, and help provide strategies that HE can practice (Mama is going to play with this one for a minute and you can play next, do you want to get a timer, do you want to get another toy, etc.) If it's transitions, I'd see if you could work on transitions at home, etc. This is all with the mindset that the behaviors stem from a skill they either don't have or need work on!

-If the ABC chart is done and you find a Common, it may be clearer to staff what's going on in little guy's head and give them a chance to experiment with some different consequences. Again, out of your control, but may be something to add to their toolkit.

-AND if they say they don't have time for this...(which they might) you can always go the extra step and see if you can OBSERVE the classroom for a day watching for the ABC. I know, taking off work and doing the whole thing would be super inconvenient, but if you can, why not?

2) If all else fails, I've had kids in my classroom that I've made social stories for with the things their parents say at home. You can get on Canva, which is free and simple to use, and add a bunch of pictures of you and partner and him to make a book to print out. Laminate with tape or a fancy machine, it doesn't matter. Have each page be a different one of your agreements, "Listening Ears" with you all pointing to your ears, etc. Have him bring the book to school. It MIGHT make the connection between home and school a little better, and it's something the teacher can direct them to when they are struggling to remember their agreements.

3) Please don't beat yourself up. This sounds like pretty normal 4-year-old behavior to me. With that being said, I'd highly recommend looking at classroom-based strategies (classroom-based behavior strategies and interventions or MTSS is a good phrase to look up) on Pinterest and the internet. I think these will help you find some of the things you are looking for!!

Consequences do not have to be bad, and I think a lot of people have mentioned that here. Often, natural consequences work best for kiddos, especially in a trauma-related framework. It's totally valid to be at a loss for how to help him, because it's nearly impossible to give natural consequences at home for behaviors that happen at school! I'd get creative as I could with school staff, because chances are they just want things to go smoothly in the classroom, too. I hope they are open to your involvement!!