Thanks for the replies, all! I agree entirely regarding not starting off with smaller DNA matches; I should note that this is most certainly a sidebar to my work on the main portion of my tree. It's only because it possibly intersects with another line of more solid research that I'm looking into it at all.
Looking further into my 12cM Ancestry match (we'll call him 'A'), GEDmatch puts our match slightly higher, but not by much: 13.7cM. I have 8 shared GED matches with A. The highest of these matches is a kit we'll call 'B'. The one-to-one tool shows me and B share 17.8cM, with A and B sharing 38.7cM, largest segment 28.7. There appear to be no matching segments that overlap between all three of us, however.
Back on Ancestry, looking through my matches, I have found two other people who appear to descend from the same family line as A. These three people all claim two shared ancestors born in the 1720s, but from there on back it gets a little messy. The trees don't agree on who the husband's father was. A claims one name, linking back to the potential Sephardic ancestor in their tree, but looking at the ages in said tree, the named ancestor would have been 14 when they fathered the son. The other matches claim another name, from another part of England, but going by the ages in that tree, this man would have been 75(!) when he fathered the son!
Looking at the above, I'm starting to think that my connection to A isn't a false match - but I think I'm going to need to do some parallel research on the trees involved before any conclusions are drawn!