No, Part 1 Reg has nothing to do with VAT paid status. It only records title and loans secured on the boat. You need the original invoice showing VAT (usually from the supplying dealer to the first owner) or a certificate from Customs confirming VAT status (usually for very old boats)
I recently pursued this issue with HM C&E and they said that they no longer issue certificates of VAT paid status and neither, they believe, do other EEC countries.
So, a complete paper trail including the original VAT invoice is needed. VAT exemption applies to boats manufactured before 1 Jan 1985 AND in use within the EEC before 31st December 1991 but papers/documents proving both these conditions are required.
Imho, assuming nothing else suspicious, that would be ok. However I would get a letter from them confirming it ie "Dear Gludy yes we confirm VAT has been paid and here is a certified-as-true copy bla bla". Then if it turns out they are wrong, you have some recourse to them.
Also get a warranty from seller that VAT has been paid. Not qualified by "So far as seller is aware". This gives you a basis to sue, and some leverage, though it is far from watertight because the seller might be had to track down. But it focusses his mind. If he refuses to warrant it, it kind of means he thinks there's a risk and -since he knows the boat - so should you think there's a risk.
Finally consider reducing the price. Consider the haircut you will take when selling if if buyers aren't as accommodating as you
I have the copy invoice with a stamp certifying the VAT has been paid and signature (that is not a copy) and can get a covering letter. Yet stric tly speaking that is not the original. I am sure that situation is not unique - surely that should be enough? - but I hear that customs officers can be very fussy on this.