OP we had this. We've got twins and an elder DS. Elder DS went to a fancy pants nursery on a lovely estate, was taught (we didn't realise how much they did until now) super table manners, many impressive life skills for a 3yo, gardening on their vegetable patch (it was also a working farm) so polite, excellent vocab, because disposable income wasn't an issue. He was sent home in his 3yrs attendance, a total of once.
DTwins go to a local nursery, much cheaper, high staff turnover and constant juggling of staff members between rooms. And one was being sent home weekly. "Hot" and yet not hot when I'd drop everything to pick him/her up and take the temperature at home. And they'd always try and get me to take the other one who was perfectly fine as well "to save me the journey later" !! It became such a problem that we bought a thermometer and I would turn up and zap said child in front of them and if it read fine I would make a point of saying I had rushed to collect a child for no reason at all, and not take them... obviously I could see the child was fine, I wouldn't leave them if I thought for a moment they were actually ill. Strangely enough, both DC who had these imaginary temperatures on a weekly basis for about three months non stop..
stopped almost overnight.
They knew I was only round the corner. They knew I'd always turn up and take at least one child, no questions asked, but they did this so often, it became clear it wasn't right. They were taking advantage. But we are generally happy with everything else, it's not a patch on DS nursery, but it's also not a patch on the price. But don't get me wrong, even with our 30hrs free childcare, we still pay £650 a month, for 4 part sessions a week, because of the consumables they charge, so it's not exactly a steal.
When they were short staffed last month, they asked me if I could not bring DTwins in for a couple of days to "help them out"!? No, if you need more staff, use an agency, and look into why none of your staff stay for more than 6mths, which would fix all this.
Take your thermometer, OP. Test in front of them. If he's not himself, or a bit sad, or a bit excitable, that's what they're paid to do, look after young children who can be superman one day and Victor Meldrew the next.
Keep records, and take pictures of DC at home, when you've had to collect him, and the temperature he registers and bring it all as a complaint if you have too. The head sounds like a bully, it's not your fault you're the first one who's stood up to this practice.