Kipper
Let's follow this through (family lawyer here and Spartacus also seems to be one).
Why don't you want her to go?
It started because he said he was a 12 week vulnerable isolater. He made all sorts of demands emanating from that about how DD (and your other DCs) and you need to live your lives in your home - eg the door handles and self distancing from all other family members, all with the sole purpose of minimising the risk to him.
This lead you to say she shouldn't go. Because it's unreasonable to expect her to distance from everyone in your home and with him she'd be cooped up and not allowed out at his home. And because of the increased risk to him and the pressure this would put on DD.
All good reasons to exercise your PR under that family court guidance, despite what the contact/residence order says, and to keep her with you.
He then backtracked. Denied being one of the 12 weekers. Told DD that they'd be going out for walks.
Even if he is a 12 weeker, you'd have to prove he was (on the balance of probabilities which you may, or may not, do). And even then he could say he's not going to do it (and nobody can make him - the risk is to him, not others). But of course that does put a terrible responsibility on DD's shoulders (although you say she is currently not particular aware of that).
So here we are now. Since he now says he isn't a 12 weeker and is not isolating, where does that leave you? What other reasons are there not to let her go? The lack of a garden is not a good one. You've said he hasn't made her anxious, so that's not a reason either. She's 9 so saying she doesn't want to go isn't enough. He can't expect to make you follow rules for a 12-weeker if he isn't abiding by them himself.
There is certainly a risk to him. DD comes from a mixed household where other DCs are moving between households. He has just been on steroids and your understanding has always been that his asthma is chronic (what evidence do you have for that? Eg past hospital treatment. But I'm not sure a risk to HIM is a reason to keep DD. The legal test is what is in HER best interests. Not his.
You do need clarity in relation to how you are conducting life in your home now that he has backtracked on his vulnerability, because the last thing you want if you let DD go is for him to keep her on the basis you've refused to follow steps to minimise the risk to him as a vulnerable person (he may yet revert to claiming he is, when that suits him - these guys can argue the sky is green and day is night one day and then the opposite the next day).
So I think now that he's denying it you need to rethink where you are going with this.
I'm not saying you are wrong, but things have developed since you first posted and decided DD should not go, and you need to look at this again from all the angles.
I will post separately about procedure and timing.