Anyone can do periodics or EICR's as they are now called regardless of quals/scheme membership. However, if you are doing commercial ones it may well be written into the companys insurance policy that the EICR must be carried out by someone who is part of an inspection scheme of some sort.
Part P.....
well this ones been done to death but here goes
Its not a qualification.
Its simply a system whereby you pay the scheme provider £400 odd quid a year to come and assess how you are running the business, check you have all the right books/certs/quals paperwork etc and look at one of your jobs.
Being part P registered allows you to self notify NOTIFIABLE works with building control for approx £5 per job. If you are not registered the charge per job would be MUCH bigger (hundreds)
Notifiable works are anything in a kitchen or special location or anything incoporating new circuits.
If I was in your situation, personally, I wouldn't bother with it. I'd just do whatever jobs came my way test them and issue a cert.
I am however not in your situation and am infact part P registered.