Surely even if the appeal is successful then she will be at least a month behind her peers already in her learning. Catching up with a months work in three subject areas is very intense, along with keeping up with the content currently being studied is going to be a very difficult task, even for able and high achieving students.
King High is really not a replacement for face to face learning in a school/college environment, so your DH is right.
If the appeal is unsuccessful then what is she going to do instead? Start at another sixth form which will take her (but she'll still be a month behind and have the same issue)? Start some vocational courses elsewhere but then be in the same situation with behind a month behind?
Surely a post-16 back up plan should have been created back in Yr 11, particularly if there had been a lot of absence or issues which could have affected the outcome of GCSEs? In my area students in Yr11 get extensive support and advice for their post-16 options and are clearly advised to have a preferred and a back up plan in place for after GCSEs so that they know exactly what they will be doing depending on the outcome of their results.
I'd be concerned that your daughter was seriously missing out on her educational opportunities as a result of this to-ing and fro-ing about a sixth form place which is unlikely to materialise.
For context, I'm a sixth form lecturer and parent of DC who did GCSEs this summer. My college wouldn't allow someone to start at least a month in and who didn't get the grades for entry or the specific grades required by the courses, even if it was only slightly off.
My college would also be very mindful of the fact that given absence and MH incidents in the past, it may well not be in that student's best interests to start a course at least a month late, with no evidence of that student having studied anything since exams in the summer (what has she been doing in the meantime since her one day at sixth form at the start of term?) and with grades which would place them at the lower end of the subject cohort.