I know that it feels like your DD is the only child with needs, but it's not the case (they wouldn't have any of those facilities if it were). And if it happens once on a particularly bad day, it'll become particularly bad everyday, as she'll know that it's something that can happen and instead of being in Reception, it'll be around all the other students. I've seen it tried and seen a child who was usually OK in a couple of minutes once their parent left massively escalating and distressing other students with SEND - and, not that you'd like this, another three students suddenly starting exactly the same behaviour after seeing it resulting in one parent coming in and more parents notifying the SENDCO that if x child can have that, they wanted it as well.
The wellbeing mentor has other children, whatever you may believe. Tying her up supervising you in school means that she cannot support the other children that definitely exist and you would have to be signed in, issued a pass, followed everywhere, watched, all to maintain the safety of other students. It would be a huge safeguarding headache that would not be acceptable to Ofsted or the LA if they were to inspect.
As they've said, it's not primary. They cannot facilitate you coming into school each day to have her with those reactions in a place that is specifically for all students that need it, not just one student whose Mum feels it is only there for her daughter's use and nobody else can possibly need it.
It's just not feasible and in all probability not remotely helpful in a secondary setting.
A part time timetable has to be agreed for a short time and a place found in AP (where you could well have exactly the same issues at drop-off). Funding from an EHCP and a place to be available would make it more likely. But then it would go back to exactly the same fulltime schooling again after a maximum of a term, usually.
The school cannot meet your daughter's needs. Hopefully an EHCP can be obtained where she can access educational provision that is suitable.