To tell exH he can’t wfh in my house while daughter is at school

Can you lock your bedroom door? If not, I'd be putting anything personal in the loft (or where he won't look - eg the laundry basket 😁) and covering my room with post-it notes saying "I'm disappointed that you are in here, but not surprised. Get out".

But other than that, it does sound like you need to put up with this briefly (sorry).

"I'll see" 🤣🤣. It's not his house or his decision. I'm assuming his arrogance is a contributing factor in you guys being divorced

I think that, under the circumstances, I’d be inclined to allow it just for the five days in two weeks, simply because of the complex arrangements and the flexibility he has available to him, which you don’t have in your own job.

However, obviously I don’t know your relationship or what he’s like as a person. If he has form for coercive controlling behaviour or violence or whatever, then of course that’s different. You, of course, understand the dynamic of your own situation.

If you definitely can’t handle him working in your home for a few days temporarily, then I think given your civility, maybe being straight with him would be a better option than just changing the WiFi password without telling him. But again, totally depends on the relationship.

So...
He was temporarily living in your home to take care of the other children for the time you were in the hospital with DD.
He is participating in the school situation on some days- picking children from your home, school visits in the middle and pick up - for the time being.

This explains why he needed (temporary) access to the key.
It doesn't sound totally unreasonable from him to want to work from your home on those days now.
It would have been be better if he asked than just assumed.

I am on the edge here. I understand you aren't happy about needing his help, and that he isn't helping you personally, he is helping DD.
Still, it might be a kind of investment in cooperative relationship to let him while the midday visit is needed.

Yeah honestly if this is short term I would just be the bigger person, put up with it and buy more sage!

Ultimately I think you need to balance up the benefits of him not being in your space against the potential negatives, i.e. your assessment of the probable response and reduction in childcare that he is doing.

What do you think he would do if you say no more WFH in your house? Is there a half way house where you permit it but email him or message that it will be a time limited arrangement with certain boundaries as you see fit and reasonable? Is this about him being in your home or a sense of loss of agency - you may be able to reclaim the latter while permitting the former.

Why is he even coming into your house? If they are old enough to have keys/get the bus then surely they can be on their own for the short period between you leaving and him collecting. Then he collects DC, takes them to school, finds a nearby coffee shop/library whatever until pickup, returns DD then repeat for later pick up, drops then off home and then leave. If it's too long to leave them, then he takes them back to his or they go into town. I really would be doing everything in my power to prevent access to the home this is not helping your boundaries.

For the sake of 5 days, can you wfh or take any parental leave? Ask a friend to help, parents? Remove his key, get a key safe and explain to the DC that this is no longer dad's home so no longer appropriate for him to be using a key. Send your message and make sure it's absolutely clear he is not to come in.

Oh yes, I don't exactly think he's a prince among men.

But, like it or not, your set up is such that he could choose not to help you. And if he did choose not to help you it would cause you a lot of problems.

He has stepped up, he is doing what he should do, so letting him WfH at your house for 5 day is the trade off.

If you say he can't WfH at yours, he might say he can't /wont cover the days, leaving you with a much bigger problem to solve...

As your daughter has been in hospital and obviously has medical needs right now, couldn’t you speak to your employer and try to take some leave because it really doesn’t sound as if you want your ex to be doing what he’s doing? You’ve said yourself that there’s only two weeks left of the school term.

While I don't think you're being unreasonable I do think you need to accept that if you say no to this he may just refuse to do it at all. Where would that leave you?
Would you be able to find another way to accommodate the appointments?
If so I would probably do that anyway, not because he shouldn't be doing this for his dd (he should) but because the less you have to rely on him the better. He isn't parenting and he isn't going to change. You relying on him just gives him power in your life. From experience the only way to avoid that is to not need him for anything.
If you couldn't manage without him doing this i probably wouldn't rock the boat.
I appreciate that you are responsible for 100% of everything and it's not fair, but you can't change him, only the amount he influences your life.

Could you just get spare keys cut for the kids and do away with the keeping a spare under a flowerpot situation? Would probably be better from a security pov as well.

It’s his child.

Just change the lock, OP.

Whilst I understand why you don't want him WfH in your house, the reality is he's being very, very flexible. Legally (not maybe morally) he doesn't need to be doing any of the child care at all on "your" days, and he's bending over backwards and being very accommodating (rightly so as a parent).

I'd let it go for the sake of 5 days. He's doing a lot to make your life easier, why not do this to make his easier in this fairly exceptional situation where despite being divorced, you're having to work as a team.