I would suggest reading through all the links and see if Budd mentioned his 'Parker Pigeon Gun'. If so, and there is no record of his ordering another, it's
likely the gun that he took to England.
I am of the opinion (and others will differ) that a cosmetic restoration will add little value, and if done incorrectly will
negatively impact value. But pride of ownership matters, so do what you want.
It certainly deserves a check, clean, functional repair and refurbish, then return to the traps!