You don't need a longer barrel and the way to get a faster twist barrel in the length that you want is to order one. The 75gn AMax will work just fine on coyotes, they're not especially tough and the AMax will yaw and fragment on impact. The 75gn Hornady BTHP is another good choice. The Berger VLDs are also worth considering.
You'll lose some velocity going with a shorter barrel, and the .22-250 is a little more sensitive to barrel length with regard to velocity than the .223 but the .22-250 will be the faster round out of every barrel length you try (within reason).
The .243 is roughly the same proportions as the .22-250 as far as the ratio of case capacity to bore size and uses powders even slower than the .22-250. Why would you think that the .243 is OK in a shorter barrel while the .22-250 is not? The .243 is the superior shooter in the wind but only when you factor in the use of 100gn+ bullets. I personally wouldn't hesitate to add the .243 to the reloading stable (it's one of my favorite rounds to reload actually), but if the OP wants to stick with the .22-250, a 20" rifle with a faster twist shooting the 70gn+ match bullets would be handy and will work about as well as you can hope for out to 400yds.
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