WIBTAH if I was to shame a ra*c*ist customer? [UK]

I (F28) work in an area where there are lots of bars, pubs & clubs around, and on the weekend we get a lot of drunk people trying to get into our business.

Today we had one (M25) stumble in, got confused, and then refused to leave.

So while we were trying to sort out an escort for him to go safely on his way, the guy just wandered out by himself, and to prevent him from coming back in, we locked the door behind him.

Our other customers can still get in and out with our supervision, just not the drunk guy, as we can simply turn him away by not allowing him entry again.

Clearly being outside must have woken something in him as he then hurled racial slurs at myself & my colleague for locking him out.

Boasting his white pride and how we were discriminating against him because he was white... despite the fact that the other customers in our venue was also like him: Young, Male and White (20/22).

When confronted about what he said, he tried to start a fistfight and had to be pulled away by his friend.

He justified himself by saying that we stole £100 from him, despite the fact that he hadn't spent anything and when he was inside he just stood there wobbling back and forth before he was asked to leave.

Now this idiot gave us his ID on entry, so we know his information alongside having footage of him on our CCTV.

Would I be the a**hole if I was to shame this idiot by sharing his picture to the other local businesses?

Among the businesses in the area, we share an online system that warns one another of potential problem customers, shoplifters, or antisocial people (etc.). This guy would likely fall under the disorderly category, so the post would be allowed, but morally, should I do it?