Can the EPS require a change in the "gender", "name" or other "proof" box in order to recognise my identity and be able to access the gender reassignment process?

No. The HPS cannot make any additional requirements except at the explicit request of the patient. In other words, it cannot rely on a change of documentation, previous diagnoses or any kind of humiliating physical "verification". Any clinical assessment should serve the purpose of health care, not to delegitimise the patient's identity.