I owned both and definately recommend the Bigma for quality on a secure platform, but hand held and tracking aircraft may be tricky and I wonder how many keepers you'll get! Muzz bought mine and maybe he has had time to use it and can offer an initial view.
The OS is a really nice lens but the images I have seen were a tad softer than those I know a bigma can take, but it is VERY close and the thing that swung it for me was the OS and hand holding. I could accept the small drop in quality as I knew I would get more keepers ... the difference is small and other owners of the OS may disagree.
The problem I had with mine was front focusing and it was a real shame as I liked everything else about it. I got a refund and opted for the 100-300mm f4 with 1.4x and I am happy with it. It is as at least as good as the Bigma, though loses maybe just 5% sharpness with the 1.4x. In some ways it would be a nice compromise in zoom: longer than the OS and as good as the Bigma, maybe even better!
Of course, the key is the OS or not ... and for handheld I just think it's probably the best all round solution. I will keep my 100-300 f4 for quality and flexibility (wildlife only for me!) and long term look at getting a fast IS prime SH to take over beyond 400mm or so and using it with my 1.4x.
There was a thread on these three lens last week at fredmiranda ... well worth checking it out and aircraft was the point if I remember! I think the Bigma won ... there was a poll too! Not certain about that tho'! Hard to know with polls if people are voting for aspitration, having owned the lens, or because they only vote for Canon L 
PS Johndm at AV Forum is an aircraft nut and sometimes visits here ... he chose the 100-400 IS. Nice choice, but problem in the UK is finding one that works nicely and isn't silly money. They seem to go for around £800-900 SH. SH Bigmas go for £400-500 and I've never seen a SH OS, but you can HK import new via some reliable ebay sellers for around £600 inclusive and with a no Duty/VAT guarenteed. I think DigitalRev offers full guarentee in UK if there is a problem.
See also here:
https://photography-on-the.net/forum/showthread.php?t=74543