Nice car. I really enjoyed my E24, despite only owning it for a short time. What can I say? I was young, impetuous and went through a lot of lovely cars (looking after them as I went!) in a fairly short period of time.
I had a quick look on eBay - I've heard some people made significant cost savings by importing tyres. It may have been viable, if looking for one or two (I'd recommend a pair), but no luck. At least, not within budget - they run to (equivalent) $150-200 per tyre.
I suspect your hope is that someone has a metric rim and tyre lurking in a shed, but checking the quality and age of that tyre would be very important.
OK. Do you have a M635CSi? I think they were the only E24 that came with the 240/45-415 (TRX) tyres on (IIRC) a 175TR415 CH TRX rim. The Motorsport (Series 2) cars had 220/55-390 tyres, but (and this may be useful) the non-Motorsport cars had 195/70-14 tyres on a 14x6.5" et22 rim.
I entirely understand your view on this. Buying a set of rims and tyres really doesn't make sense if you're only looking for one tyre, but in this case it appears that a secondhand, decent set of rims (with good tyres) will cost less than just one metric tyre. This would keep you on the road, and give a little time to find that elusive correct-fitment tyre. (I had a similar issue with a Porsche 928: 245/45-16 tyres were completely unavailable for a time, and owners were forced to wait until a manufacturer produced a run of them.) There is an additional issue in that it leaves you with a set of rims to store, of course.
The E34 had 7x15 et20 rims which are similar in appearance to teh stock E24 rim, and would fit easily. They're not expensive, although I don't know of any currently for sale, but would bolt straight on and not significantly change the appearance of your car. They should be easy to resell once you have the correct tyre for your metric rim, too.
My apologies - i appreciate I'm only really reiterating what others have said. I think we'd all like to see your car and if there was an easy answer to finding a good metric tyre, that wold be the choice.