Yes you can do it in a permit aircraft, so long as the training organisation / instructor and examiner are happy with the aircraft. You can't rent the aeroplane, but if you are sole owner it's no problem - I'd need to double check the current ANO / exemption provisions if you're a partial owner, but I think that's possible also and somebody might know the answer without me having to dig.
You need training as required, and the instructor to be happy for you to go forward for test.
The test will be essentially the same as your original PPL skill test, except for a bit less nav.
G