It's horrible that your mil has terminal cancer but it doesn't mean that she gets to demand her very young gd comes to stay every week!
When they ring up this Thursday, say she's not coming this week as we have other plans. Make something up and do it if needs be.
In the same conversation say 'but we (ie all of you) could come to visit for a couple of hours (or however long you want) on Sunday instead (or whenever is good for you) so that you're not cutting off contact completely.
And make sure you are out of the house when they say they will call around if they insist they are still taking her. If they are still there when you get back then be very surprised and say 'But you knew we were going to be out. You knew we said she wasn't coming to you. Doesn't matter how long you stay there for, she is not leaving her home today and she is not coming to stay with you. If they try to persuade you some more, then actually tell them to stop being ridiculous and stop bullying you, she is your dd and she is staying with you. final. Sometimes telling somebody this will stop them because they will then start to try to justify that they are not being ridiculous or bullying but you can point out they are repeatedly trying to make you do something unreasonable, how is that not bullying. Difficult I know but try to match FIL confidence in himself with confidence in yourselves that you are right - you might have to think through everything he would say and then how you would react in advance, so that when you do talk to him you are prepared and don't end up giving in.
They know that they have a good trump card to guilt you by saying that mil is terminally ill but this doesn't mean that they get to have your dd - just means that you try to visit them a bit more often.
You could try saying that you can't cope without having her at home, that it's too much for her to be away so young.
Or you could be brutally honest and say that she has been ill every time she comes back from you and I'm not prepared to do that to her any more. we'll all visit together or nothing.
It would be different if she was staying there for an occasional hour or two while you went to the doctor/shops/hairdresser/sleep/etc - a few sweetie treats from grandpa and a nappy left on a bit long wouldn't make much of a difference. But that's not the case. And you are seeing time and time again that staying with the PIL is causing problems with for child.
And now you have taken her to the gp so they will start to see a pattern too if you keep taking her in after repeated trips to PIL and they might start to ask why are you letting her go if this keeps happening? So tell them that for the while the doctor has advised that dd should stay with you and that she is too young to be away on her own etc etc (tweak to make it as strong as you want), particularly if they are likely to believe you have to follow doctor's orders. And if they don't then - works both ways - they are telling you to ignore the doctor's orders and let her stay with them - why would you ignore the doctor's orders in this instance when you want to follow them?
Just out of interest - did your mil hand your dh over to her PIL every week when he was so young? Do you know why she feels such a strong entitlement to have so much access to your child?
Good luck - take strength from the support of MN - I don't think you are going to find anyone on here that thinks that you are being unreasonable by not sending your dd to your PIL just because they want her. In fact - ask - friends, family, MN _- I bet you will find very few people who would send their dd in such a situation. So be strong for your dd and stand up for her.