Thank you for the advice, everyone.
I have hardly ever smoked it, so smoking it a bit more would probably help.
I am very wary of sanding pipes. For reasons I cannot recall, I decided to sand the varnish off my brand new Dr Plumb 5031 (?) around 27 years ago to make it look more natural. (OK, I was young and I was even sillier than I am now.) In doing so, I ruined the finish and obliterated the markings, so I can't be certain whether it even is a 5031 and not a 5081. But it has slowly darkened again over the years and it is probably now my favourite pipe.
But back to my Dad's Criterion 164 with the scratched bowl, while I might try the beeswax, I don't want to do anything that would ruin the character. My Dad has been dead for 18 years now. I had his Zippo, and it got stolen; and I had his watch, but it went missing while getting repaired; so to have his pipe is quite special.