Having ridden about 15,000 on a barrel that has a gudgeon groove down the side about 5mm deep, go ride it.
It will loose a bit of compression on the bad side and when running wide open throttle the breather will become an oiler.
Apart from that enjoy your bike and save up for a set of oversize pistons.
These are BSA's and they will bloody well run with a lot more damage than that.
If a bit of smoke & oil worries you then buy a Honda.
Back in the 60's we ran on a lot worse that that.
Had a riding mate who rode to uni for 3 years on one cylinder cause he was broke and could not afford a new set of barrels so he took the adjusters out and put a PCV valve in the plug hole.
It was a touch on the jerky side in heavy traffic and would have made embarrassing noises if the mufflers had any baffels in them, but apart from that it got him to UNI 4 days a week, out with us on the weekend, down to the pub Friday & Saturday nights , to footy on Sunday then back to Coledale Leagues club Sun evening then back to uni next week. And a lot of this was 2 up
Mind you we were a touch on the lighter side in those days.
It seems to me that the bulk of BSA owners seem fixated on reasons not to ride their bikes rather than getting out there & enjoying them.