I believed there was still a 21 day time limit to challenge the NoE regardless of payment.
There's no time limit to submit to TEC (and therefore TfL)?
Probably best you listen to us then; stop making up your own rules.
I'm not up to date on TEC current processing performance but I doubt it has improved much.
Hence there is no chance of you meeting your imagined deadline.
The first three posts here were to prevent you being subject to any deadline or rush.
Your daughter must attend to swear the forms to get them witnessed; it is not something you can do on her behalf. Anyway it takes about two minutes.