"I was told at beginning of year that SA+ should be fine, but I'm not entirely sure what my DD is getting in practice to support her. There are 2 TAs across 3 classes, and that is all the support that she can access at the moment. The SENCO mentioned statementing in passing a couple of weeks ago, but today was supposedly supposed to go through his reasons in more detail, although he spoke briefly to me yesterday and said that her results were too high to take it forward (she's just done her KS1 sats)".
School Action Plus can promise very much but in practice does very little. I have heard all this said before about "SATS results being too good" - its all nonsense and designed to put parents off from applying, the very parents whose children need Statements.
"I don't really know much about statementing, so sorry if this question is dumb but are you saying that if a parent applies then they can appeal if it is rejected, but if the school applies and it is rejected, they can't appeal?"
Exactly. Arm yourself too with knowledge of the Statementing process, knowledge is power. The SEN COP (code of practice) should be online.
"Can the school help me put forward an application but in my name?".
Would not do this. You're far better off personally applying without their direct involvement initially. You would need to write to the Chief Education Officer at your LEA and give them six weeks to reply (mark that date on our calendar). Schools can also sit on such apps for months, at least if you personally apply you know its been done then.