Well, if he was screaming I'd probably hold him rather than put him down in the crib. I wouldn't let him be somewhere where he is likely to bang his head either. I'd hold him firmly if that's what it took to prevent him hurting himself, firmly and with gentle words to reassure him. That doesn't mean he is going to stop screaming but it does mean he knows he has your support and care. Sometimes our support, cuddles, words, might appear to be ineffectual but that is not so, your DS knows you are there with him even if he doesn't stop shouting or crying.
There are 2 ways forward with this.
One is to just cold turkey stop the night feeds. You must stay with him and help him with his upset and frustration throughout though and it sounds like it is going to be hard work. But on the other side he is likely to stop expecting or asking for night feeds within about 3 nights, possibly a week. The thing is though, he is going to protest, you must expect this, and you must be ready to stay calm, peaceful and supportive and to NOT give in and BF after a while otherwise it undoes all of the progress and only teaches him that he needs to cry and shout for longer to get it. It is not fair on him to confuse him by denying him the BF and then giving it.
The other is to do more of a gentle wean. Have you read Pantely's Gentle removal plan? Here's the link
Gentle Removal PlanWhilst your LO is older than the method is described for I see no reason why you couldn't use it as a basis to wean him off the BF in a non-screaming way. You would adapt it to suit your particular situation. The main point of the method is to keep moving forward with the wean, each point gets closer to the goal.
I imagine the gentle wean taking longer as it is a slower step by step approach.
It depends what you think suits you best really. Cold turkey and likely faster or gentle wean and likely slower. either way you need to be focused and determined.
Can I just check, is he getting enough milk in the day? I'm not talking of set amounts really more that if he feels he needs more milk/dairy in his diet perhaps this could be contributing to his demands at night? You mentioned he has transitioned to cows milk at day care, is it a reasonable portion he takes or just a few sips?