School are being bloody ridiculous! They are not medically qualified to make the judgment re your dd NOR to put others which could include immunocompromised or with conditions that infections can be worse than for others at unnecessary risk of infection!
I'd tell them where to go!
Re dds illness though, how old is she? Has all the usual suspects for her age been ruled out? I'll be honest I'd be thinking something like glandular fever may be the culprit.
"Can you get a GP to sign her off? (No idea if this is even still possible!)" putting unnecessary extra pressure on the Nhs is unacceptable too.
I had similar issues with dd when she was in hospital for a couple of weeks with an infection (her disability - which they were well and repeatedly informed about) means she is more vulnerable to certain types of infections and they are more debilitating/serious than they would be for people without her condition. She'd also lost a significant amount of weight as a result and was very run down and frankly knackered!!
She got out of HOSPITAL on the Thursday, I called to update them on the Friday, they seemed understanding/sympathetic...then on the Monday I got a call asking where she was!!
We'd been advised to keep her off AT LEAST another 1-2 weeks and luckily that and other advice was in writing. I made it VERY clear I would not be sending her back until she was fully recovered. The teacher we'd been dealing with had been (until this point) to me polite, understanding & sympathetic. Dd had been saying that wasn't how she was with her but of course sometimes kids & teachers clash so I had put it down to that. This day the teacher lost the temper and swore at me, the mask slipped! From that point I insisted that dd have a different teacher for pastoral care and I refused to speak to her, her attitude was atrocious.
They were still complete arses about it. Long story short dd (and I) felt completely unsupported by the school and she ended up leaving school younger than she would have if they'd displayed an ounce of compassion or common sense!
Fwiw she's a bright, hard worker and had gained exam passes a year ahead in several subjects and passed the others with good grades before leaving, she was on the honour roll and in top 2% for achievement before leaving and has gained further qualifications since leaving. So to my mind they also shot themselves in the foot by basically "managing out" a good high achieving student rather than helping her get back on track.
Get everything in writing and back yourself up, but stand firm. You know your child & their health best.
The more I read/hear about how schools are behaving now I too am unsurprised home education is becoming so popular.
"Day off for a blood test? Come on." Depends on what the blood test is for, where op & dd had to go for it, if there were other considerations around how it was done eg some blood tests aren't done at local health centres or hospitals, the patient may need to fast or intake something to facilitate the test...
Eg One of the "just a blood tests" I once had to have done meant a trip to our nearest city which is a good distance away especially by public transport and I had to have a fasting draw done first, then given something to test how my body responded to that and more bloods drawn at various time periods after the other substance was introduced to see basically how well I metabolised that. That certainly took a full day!
Even relatively straightforward ones clinics are often running late, it can be difficult to draw blood from children etc.
"Check that with your school and maybe make appts for around 11am in future, so that your DD is marked present before you take her out for her appt." yes cos it's that easy to get appointments at a time that suits when it's virtually impossible to get appointments in a reasonable time scale at all 🤔