For me, the key is to use two bore snakes. One wet and one dry. I soak the bore first. If it's a shotgun, I soak it with MPro 7 -- just spray it directly in the breech (never contacting wood). A rifle of mine is likely only to have 50 rounds of copper-jacketed through it between cleanings so it may just get soaked in Breakfree (doesn't remove oil like Mpro 7). When the cleaner is running out the muzzle, I soak the first boresnake (Hoppes Viper) with the same cleaner. I don't totally saturate it, but it's wet. I don't want the cleaner to gush out when it's squeezed into the bore, but it will be wet going through it. A few passes and it should be fairly clean. Then I take the dry bore snake and run it through. If I used MPro 7, the dry bore snake will have some oil on the end to coat the bore, especially if I'm putting the gun away.
The wet bore snake gets most of the fouling and has to be changed more often, but all my handloads are very clean shooting and they last me plenty long. I have not used a copper solvent on a long gun yet. I suppose I could use it with a bore snake. Isn't the ammonia actually a gas? I believe it's a gas that's dissolved in the liquid. So if the snake is soaked in copper solvent, the ammonia should all evaporate and go away after a while. I suppose the other thing I could do would be to use a third boresnake for copper solvent so I'm not mixing a lot of chemicals.
I have a bore snake for my handgun(s) but I prefer to use a collection of brass Dewey rods. For one handgun, I have a half dozen of them with different attachments (chamber brush, bore brush, jag, loop, mop, etc.) That way I don't have to change attachments. With the short barrel and revolver chambers, the rod is faster than threading a boresnake through each one.