For LSE that would make sense, since they weight the personal statement more than any other uni. If I'm being honest, I wouldn't apply to LSE at all without FM regardless (same goes for Cambridge). For other unis though, not having FM shouldn't be a problem. I applied to econ at uni this year without FM and it hasn't limited me, I just had to avoid Oxbridge/LSE. My point about the TMUA was that unis that don't require it don't get sent the score, so it would make no difference to her application whether she got a 9.0 or a 1.0 (i.e. it wouldn't change her chances at all).
I do agree with you though, it's good to show extra maths knowledge, but how important that is depends on what unis she's applying to. Some unis like Bristol don't even read personal statements other like LSE weight it very heavily. The OP mentions "good universities" which is quite vague. I assumed they were referring to most RGs/top non-RGs, you seem to be referring to mainly Oxbridge/LSE so I guess we interpreted it differently.
I think it would be helpful to know which unis OP's daughter actually wants to apply to.