"or other attributes of sex"
Currently the only proof of "living in the acquired gender" that is required for a GRC (in addition to the diagnosis of gender dysphoria) is documentation, eg. of change of name and/or Title on household bills.
It is hard to imagine that the PC of GR under the Equalities Act could require more evidence than is required to obtain a GRC under the Gender Recognition Act.
For example, under the current Equalities Act system, all a person would have to do to be covered by the PC of GR is to state that they, for example, intend to ask their Utilities Company, etc. to change the name on their account.
Although I doubt that they would even need be that specific. Probably just,
"Hey boss! From the start of next week I intend to be LiViNg As A WoMaN!"
"OK. Contact HR and give them your new name and they will change your Employment Record. Anything else we can do to help you with this?"
Under the more restricted system you propose, all they would need to do is state that they have changed the name on their Gas Bills, etc.
Or just more vaguely state that they are NoW LiViNg As A WoMaN.
Or man, as might be the case.