No. You would need a DClinPsy or EdPsy to gain HCPC registration. No one will hire you otherwise, and you would also be putting yourself at risk of using a protected title.
Beyond that the neuropsychology training route would build on the material introduced in the doctorate, and formal registration with the Division of Clinical Neuropsychology would require you to be working post qualification in the setting already with clinical supervision. It's those people who would be eligible for the course that would allow them to practice clinically, and those courses are often sponsored in Band 7 roles where clinicians are skilling up to do a role.
Those courses usually have a 'sister' course open that even shares the academic material, to non-qualified applicants out of academic interest, or international students who have different requirements in their native countries. They are clear that won't make you qualified as the majority of learning is still done on the job or with your supervisor, with the academic element as 'a skeleton' you hang it off.