I think Omega is about the only replacement system available for the electronic ignition bikes, it uses the ignition coil that's installed, and some wiring in the bike, all of the other components are from Omega.
I know a member here, Semper Gumby, installed this system on his '80 R65, may want to send him a Personal Message for feedback on his experience, but from what I can remember, he is quite pleased with the Omega system.
How long do you let the bike run before shutting it down ?
I don't know how much more time you want to put into this problem, but the ignition canister under the front cover of the engine, hasn't been addressed yet.
There's two allen head bolts that hold the front cover on, ( before you remove this cover, remove the positive lead from the battery, as there some connections in there that are 'hot', they have power all the time, so you could end up with a 'shower of sparks' if you touched the cover to one of them) then I think there are two phillips head screws holding the cover on the canister.
You could try spray freeze on the the Hall Effect sensor area to see if it is the faulty component.
I don't think cooling the outer housing will do you much good.
Other than this, the only things left in the system are wires, ignition switch, and tachometer.
Gets a bit frustrating trying to find an intermittent electrical problem, then usually it's not a complicated repair after the problem area is found.