AtoA flights in the UK, no more than 50 miles from take-off point can be done with a CPL, but you'll need an "A to A AOC", which is specifically designed for the purpose.
Getting the restricted AOC will cost you the greater of £288 or £88/month applied for, plus quite a lot of paperwork.
All the application details are on CAA's website.
Letting them handle the controls can be authorised in your AOC ops manual as "special category passengers" but CAA may not accept it if you seem to be doing what is effectively flying instruction.
Yes, the aeroplane needs a CofA, maintained to public transport standards, the same as it would in say a flying school.
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