Martin, you say 'I'm sure you want to know just as much as you want your cousin to know, but just settle for finding out for yourself'.
The only problem with this would appear to be that the 'conundrum' might be difficult to solve unless both parties co-operate.
I know where I connect with my closest matches, apart from my 5th and 7th matches, managed by the same person at Ancestry. Using the shared match feature I know the probable line. They would appear to be 3rd or 4th cousins, so not as close as your unidentified match, Kathy.
This match has replied to me but has no tree at ancestry that I know of. She has told me that there is no time or place where we could possibly be connected. When I suggested that it was with this particular line, the suggestion was again refuted - apart from a couple of illegitamicies, their tree was well researched, there was nothing to prove paternity was from a roving raiwayman (the family I think they are connected to travelled widely over northern England in the mid to late 19 C, working on the railways).
I would think that this was either one of her illegitamicies, or a NPE which she knows nothing about, and it was indeed a roving railwayman responsible. Either way, the conversation is closed, no further responses.
So, though I would dearly love to know how these two are connected to me, looks like I won't find out any more.
Regards Margaret