Sadly, some responses you’ve received are really quite dismissive of what severe anxiety may mean or how crippling it can be, and especially if it’s something they’ve suffered with since the age of 7. With no understanding of how it affects them, then some of the comments may be quite unfair.
Things we had to think about, which are possibly the things you are also, quite reasonably, worried about.
How stable is their mental health at the moment?
When in an anxiety crisis can/do they initiate getting external help for themselves (or for example do they become non verbal or agoraphobic)?
Can they manage the journey alone on public transport if they need to come home? Some anxiety disorders may mean this isn’t possible. Are you able to get to them easily if required (can you drive, have access to transport)?
Are they currently receiving treatment/CAMHS support (as them changing health authorities may impact this or waiting list positions etc)?
How can the support they need (currently being provided by you, their college, local services) be replicated using other services/providers? There is DSA and PIP to look into as well as pastoral support.
If this is all good, then Bath Spa or a London Uni may be a great option for them.
There is also the possibility of waiting a year and reapplying if there is something that can be done that may make them a more attractive applicant for their preferred local course. Or look at other local providers for similar or subject adjacent options, or consider the OU. Or using the year to develop the skills and strategies for living further away, by deferring entry into a known Uni.
Not all DC are the same, or ready at the same age, or even going through the same things, so they should feel free to do what’s right for them.
Wishing you all the best.