This post has raised many issues.
Gnomon, It should not be difficult but is may be (relatively) expensive, depending on the age of the weapon and who you ask to do it.
Diggory (Smallbore) and I offer relatively cheap transport of pre 1899 shotguns to the USA if they can be accommodated with our regular trips out to attend shows or whatever. Post 1898, delivery deadlines and other complications can make this much more complicated and expensive. (Diggory may have other matters taking up his time nowadays!)
Other shippers specialise in more modern weapons and so may be (relatively) cheap for these. I would recommend you contacting Dig or I for quotes/recommendations.
Jim, The auction system in the UK has increasingly been treated as a 'retail shop' by the internet buying community, much to the UK restorers' frustration.
Many guns achieve prices way beyond what they should because bidders choose to forget that they are buying at auction, where (other than the proof laws) 'caveat emptor' generally applies.
I spend at least 45 minutes per gun closely examining and measuring before carefully calculating restoration cost, retail price and hence the bid I can make.
Up until the last sale, I have averaged 1 gun purchased per sale at Holts, Bonhams and Gavin Gardiner. At provincial sales, where significantly less information is provided to purchasers, I have only bought a total 2 guns in recent years as every gun has been purchased for ludicrous prices, given their wall thicknesses etc.
This is all to do with the prices that these guns are achieving, if the price is right virtually any gun is worth restoring.
I can honestly say that I have only 3 times in the last 10 years discovered a gun at any of the London gun auctions that was out of proof and all of these have apparently been genuine mistakes.
However, when it comes to a non-proof related condition, pretty much anything goes. I have seen many Big Name guns being sold with no wall thicknesses published - caveat emptor.
All I can say is, please stop expecting to pick up a bargain from the big auction houses. These guns will have been crawled over by the likes of myself and Diggory and will most likely have issues that the auction house are unaware of. That is why we drop out of the bidding.
A common belief is that wall thickness is a factor in proof. It is not. I have had guns proofed down to 0.014" MWT (for a family heirloom).
Cracks in the action are a proof issue, not the forend.
Loose lumps are a proof issue but unless they cause the barrels to come off the face, they are very unlikely to be picked up at proof and no auction house has the time nor facilities to check them.
I haven't been aware of any caustic bluing of soft solder barrels in the UK but I suppose it may happen. I would guess that this would only become apparent when the barrels were hot tank tested for rib leaks. Perhaps this shows my ignorance or lack of exposure. Auction houses can hardly be held responsible for the poor practices of their vendors. Or perhaps they should be, another interesting discussion there.