Is it possible to ask the chap who first posted the picture?
Dave, the pistols are in the National Maritime Museum in Greenwich UK. The photo and description appear on their website, which also says the guns are not currently on display.
BTW, I posted a query on the ML Association of Great Britain's forum, thinking a more authoritative answer on the holes might come from the other side of the pond, and 'fishing weights' was the only practical answer anyone could suggest (but doesn't seem likely to me that a Royal Navy Captain would keep fishing weights in a cased pistol set that eventually ended up in a museum).
BTW, the Museum description lists the guns as .75 caliber. One response from MLAGB was that sounded unusually large. I pulled the photo into photoshop and did some measuring, and either the pistols are about 20" long, or those balls are only about 50 cal. Maybe they're .57 cal, and somebody transposed the digits.
I found an email address for the Museum for inquiries and sent the question to them. I'm going to be surprised if I get much of an answer, but it won't hurt to try.
SCL