I wasn't really thinking of a professional copying service. There may not be any, partly for the reasons you mentioned before.
I was thinking of some nearby audiophile with a decent rig, willing to make copies for a reasonable fee. You might ask on Vinyl Asylum. Copying to digital is something several members do there regularly. Search for posts by John Elison. He's the acknowledged expert. They may even have DIY solutions that would fit your needs and budget.