I'm struggling to find a place to allow me to sit CPL exams while I wait for good weather to qualify for my licence. I'm just looking to save a few months which should be possible surely.
Have you explored Commercial distance learning courses a little further afield than UK SE?
If you meet the minimum requirements to start I can't see why any school would put you off. Ideal time to do it. Or are you wanting to start the CPL before completing the PPL, in which case....