You will have to pay the CAA for the conversion and inspection of your machine (about £400 IIRC).
Insurance should make no difference unless the use is going to change (as in putting it on a PT C of A to do aerial work, hire or training on it)
The usual biggy is that engines on aircraft on Pvt CofAs can have an engine running on condition which is something you can not do for PT CofA when you can only run it to TBO and an extension of 10% (ISTR you can apply for that twice)
HTH