This is more difficult than usual because of the ongoing disputes between Qatar and its neighbors -- if this were a few months ago you could have flown Saudia, though whether you should want to is a separate matter. Similarly you could have flown AF or KL into Dubai and taken a short connecting hop on a local airline the rest of the way, but that option is also no longer available.

I'd recommend you suck it up and fly QR. An hour on an AA flight to JFK or ORD won't kill you, and it is far better than creating some elaborate multi-stop journey through some unpleasant airports just for the MQMs. And Qatar is generally though to be a pretty good airline, especially if you will be traveling up front.

On SkyTeam, other than the MEA option, which may be difficult to piece together, the other most logical regional airports that you could fly AF or KL to (and then book a separate non-SkyTeam final leg from there) would be IST or KWI, neither of which is a hugely pleasant place to be making connections from, and that final leg from IST would actually be pretty long. But maybe KL to KWI is a winner if you really, really value the MQMs?