Goex...

Here's a thought... and I'm gonna assume that the pins are somewhere around 3/32" dia. (maybe larger) so if you have a machinist friend, remove the pins as best you can starting on the left side and tapping them toward the right side. Color code them as to their place in the barrel. It can be as simple as using a "sharpie" or paint stick to make a single mark on the one closest to the muzzle and two marks on the next and so on, I'm sure you understand. That way, they'll go back in the hole they came out of.

Next, have your friend put each pin in the appropriate size collet in a tool room lathe and simply dimple the face on each end with a center drill or small drill so that the punch you're using to remove them won't slip off center. I'm not used to dealing with pinned barrels, but if you feel you need to remove the barrel this may help avoid damaging the wood each time you choose to remove it.

I believe most pins (at least in the manufacturing world) have a slight taper and are meant to be drifted from right to left directionally... meaning put them back in from the right toward the left with the smaller (left) end being inserted first, and remove them vice versa. I would also mic the pins at each end after you remove them and see if they are indeed tapered. If they are, mark the smaller dia. end as the end to be inserted first or have your machinist friend put a bevel (chamfer) on the smaller end to aid in replacement and help identify which end it is.

At least that's how I would do things if I was still in the machinist's world.

BPS