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

i have a jeager rifle with pinned barrel i just put a piece of tape with a hole in it over the pin and the key for me is to have plenty of light,use a magnifier i also have a 1-1/2" thick piece of foam i lay the gun on flat on it's side which helps control the "hit" of the punch the foam also allows the pin to be pushed out of the stock. i also push down on the stock so there's no bounce when i hit the pins. i think it's much easier than trying to hit the pin out sideways say if the gun is in a vise. so far i have not damaged the wood but do not like taking the pins out too much due to the fact that eventually the holes will get sloppy.

On my Traditions, I use a simple punch and push inward by hand and they pop out enough to either pull out by fingers or with a small pair of pliers.

That would be expensive to replace.
I've never had the pins out of my little longrifle for that reason. For cleaning I've always just pulled the flash hole liner and used a flush tube. Has worked well for years.

as far as cleaning i just put a rag on the nipple let the hammer down and put some of my cleaner down the barrel to soak for a few minutes after a shoot then i just take the range rod with a patch and push it down the barrel a little bit pull the hammer off the nipple then push the range rod down the barrel the rest of the way it's like a power flush and of course i tilt the rifle as to not get the dirty fluid in my lock.

mine are way too tight to push out like that.

I tried that stuff and although it worked OK I prefer my previously mentioned Turkey Tracks. Another good product is the citrus based BP solvent at Cabelas. I buy this stuff by the gallon. I don't see it listed on their site but my local Cabelas carries it.
I'm sure a home made recipe would work nicely. Maybe a combo of lemon dawn, lemon juice and water would make a great cleaner?
Follow that with dry patches, alcohol wipe and dry and a couple oil patches and you should be good.

That's a good idea. Maybe even something better than the tape would be a small wood jig (like a popsickle stick) with a hole in it to place over the pin and tape in place.
The pins on my rifle were small. I used a 1/16" punch to tap them out. but I only did that once because I had to get into the ramrod channel to remove a stuck jag the previous owner lost in there. I got the jag out by using a dremel tool to cut some of the barrel channel out to get to the ramrod hole.

yeah know that i really give it some thought you can make a little jig that holds the punch perfectly over the pin and tape it on so you don't have to hold it at the same time? i'll play around in the shop this week see what i come up with let me know if you figure something out ?

That's why I suggested "dimpling" the face of the pins once they're removed, so he won't be so likely to slip with a punch and damage the wood in the future if he intends on removing the barrel again.

BPS