Absolutely agree profchris.
AND Andy, they may not have used the same terminology then as we do now, but today we'd call this instrument a LRU ("Line Replaceable Unit"). Therefore if that particular instrument went unserviceable while fitted to one particular aircraft it would have been removed and replaced with another one from stores (same P/No of course). Meanwhile the busted one would go to the instruments workshop (or possibly back to the instrument manufacturer).
Once repaired and tested out OK, it would then be sent to stores (complete with a serviceable tag), ready for the next time that instrument broke down.
So if the squadron to which the aircraft had been originally fitted had, say, 20 aircraft in total, you can easily see that the next time a Beam Approach Indicator broke down and needed replacing, it would be, in this example, a 20:1 chance that it would go back onto the same aircraft to which it was originally fitted when that aircraft left the factory new.
To further compound your difficulty Andy, if, as appears likely, this type of instrument was used on other aircraft types as well as Lancasters, maybe only Avro types, but possibly types from other manufacturers as well, then your chances become even more mathematically difficult (I'd say "impossible" actually.
The only possible route out of your problem I can think of was IF this type of instrument was always accompanied by it's own Log Book. (I say IF 'cos I don't know, but think it highly unlikely, because back in WWII days and into the 1950s & 60s, it was unusual for relatively small/cheap/simple items such as this instrument to have their own log books. AFAIK own Log Books were normally reserved for major items such as aero engines , landing gear leg assemblies, etc). But if it was the practice then to have individual Log Books - and IF you can find the Log Book that belongs to that particular instrument! - then the Log Book should list the dates on and off and details of every aircraft to which it was ever fitted (and a fair bit of other data too).
But old though I now am, I only started my RAF training back in 1961, so personally I just don't know what the documentation standards for such instruments would have been back in WWII.
As said at the beginning of this thread my own guess is that it will be a matter of pure luck, and only luck, if you can solve your puzzle mate.
But anyway, good luck