Firstly, I think you need to
Firstly, I think you need to ensure that your Isurance covers you for European travel longer than a month or so.
I can’t imagine why you would need to re-register your vehicle if you move from one country to the next.
As there is no entry/exit visas needed from EU countries to reciprocating countries, there really is no record of if the vehicle has stayed continuously in 1 country or been goin in and out. Just ensure the vehicle is properly insured, taxed and tested inline with it’s country of origin, ie UK and there will be no hassle.