"but I don't want him to be stuck with a SEN label for the rest of his daysl."
Well for starters educational labels only apply while he's in education and needs support, so if it turns out that he does settle in and there comes a point where he doesn't need support there's no need for the SEN label anymore anyway. If it doesn't need to be a lifelong label, it won't be.
"Will it help him to be told he has a condition, or will it make him feel more excluded, more 'other'"
My son has Asperger's, he's bright and he could tell (at an older age than your son) that he was struggling to do things that other children did easily and without even thinking about. His Asperger's made him feel excluded and what has helped with that is the diagnosis, the knowledge that he is not 'useless' (his word, not mine) but different and that there are other people who do feel similar to him - that he may not be like other children in his class, but that he's just wired a bit different and that he's 'normal' for him.
"I just want him to be happy and I'm devastated to think of him feeling lonely and isolated or worst of all unhappy."
My son's happy, he'll be 15 in a couple of months, he's doing pretty well at school (bar a few minor issues) He's never going to be Mr popularity, but he's got friends...I worry about him, but he doesn't, it affects me much more than him, which I figure is good, it's just me being a worrywart then isn't it, lol.
5's a fairly common age to spot Asperger's ,it's subtler than other things and it often isn't noticeable until they're in school, my son was 7. I was chatting once to a parent support worker once about feeling bad about it being so late before I picked up on it, she laughed and said that she'd worked most of her adult life with people with ASDs, had a daughter who was autistic and didn't notice that her son had Asperger's till the school suggested it...so don't beat yourself up over that at all.
and crying at the teacher, dear god if I had a pound for every time I'd done that, I'd be loaded by now, lol, of course you were upset, it's an upsetting thing.