The Palermo lounge isn't terribly good, it's just a coffee machine and a few armchairs landside, albeit with a fast track passage connected, it's scarcely better than the main terminal, and it's the only lounge there. It's not unusual for a route like that not to have a lounge, and it's not listed on the BA website link to the oneworld lounge directory. Nor is it in this forum's database either, it is identified as a lounge less location. It's a fairly small airport, and probably not worth arriving at too much before departure. How you could spend 30 minutes finding it when the two landside floors are about 200 metres long will remain a mystery!

What I think has happened with your boarding passes to EDI is that PMO are not on a full FLY implementation, but a cut down version, and so indeed it is possible you could not check you in for EDI. There are only about a dozen such locations, mainly in Greece, PMO I know to be another, it's a fairly new route, though I notice it wasn't culled like some other locations were a few weeks ago (e.g. MPL and BIQ). Using the BA app you could have done so about 10 minutes after the PMO check-in process, but I know that is not intuitive. You could also have tried that waiting for the bus connection. You were 5 minutes over the minimum connection time, so it was always a fairly tight connection, and at that hour it doesn't take much for the connection to be put in jeopardy. Sometimes you win, sometimes you lose.

Now for the Involuntary Denied Boarding aspect: you have an arguable case, and I'd refer you to the main EC261 thread for the details of that. There is case law which supports your position. BA will also have an arguable case since there is material in the Regulation which supports their decision to not get you on the flight.

You also used - I'm sure deliberately - a significant phrase: "not having taking reasonable skill and care during the check-in process at PMO?". This wording is lifted from the Consumer Rights Act 2015 (and previous consumer legislation on services dating back to the 1980s). You may have a case there, a thread on this legislation is by coincidence about to be launched in this forum in the next day or two. On the face of it, you may have a case here, certainly an argument.

In short, the lounge you can forget about, for IDB you may have a case under EC261 but it's not an overwhelming one, for the check-in failure, well that may be your stronger argument under the CRA.