One quirk on my '64 500/1 is the slightly misplaced tailpiece maybe, a rush job at the factory trying to get the order out who knows. I have never thought about trying to remove the 2 nails & strap pin, and relining it but here's my thoughts on doing it.
I'm nervous about snapping one of the nails as they've been in there almost 60 years, and if they did come out easily would there be enough of a wood block to tap them back in?
Should I just leave it alone, what are your thoughts ?,thanks in advance.
View attachment 5260336 View attachment 5260337
Are the strings centered correctly on the neck with the bridge in that position?
When was the last neck reset?
The tailpiece & nut position are the two variables. The neck could also be very slightly off set (downwards or "headstock to the right" in the picture perspective) and can be dealt with at the next reset if it is. A luthier should be able to sort it the next time it is due.
I do see your bridge PU is also shifted flush with the left side of the surround with a larger gap on the right..
I'd check that the strings are centered on the neck first & make sure the bridge is set for that variable, then get the PUs adjusted in the surrounds (the poles do not have to be exactly centered under each string - a Höfner Toaster had 5 pole pieces... align that

) and then see if the neck is correctly set, then consider the tailpiece move as an option. A luthier will be able measure the center point exactly and see what is off (neck or tailpiece). Personally, on a 1964 500/1 I'd leave the job to a (trusted) luthier when you take it in for the next re-set.
If the offset is slight and your bridge is shifting, you can rub a bit of violin bow rosin on the base of the bridge. Doesn't "glue it", but stops it from sliding to the right until you have it sorted. I do this on all non pinned floating bridges.
=> "slightly misplaced tailpiece"
It's not that uncommon, I have shifted tailpieces and Bigsbys on a few instruments (e.g. Gibsons & Gretsches), be aware you will probably see a difference in wood color due to the age of yours around the anchor plate if you do.
Never tried pulling the two Höfner tailpiece nails, so no experience... is fine on my '63 Club 60 & close enough on my HCT 500/1.
=> "would there be enough of a wood block to tap them back in?"
See 02:50 on wards: