Yes, not awake yet, but what you say fits well. You could largely avoid the whole recoil issue with goggles, an IR light and IR laser.
You could go cheaper and a little heavier with a helmet mounted Sionyx digital night vision (DNV) setup for one eye or spend more and go with real traditional NV, Gen II+ or Gen III. Other than the Sionyx, I don't know of any DNV that is small enough to helmet mount. The Sionyx stuff is cool because you get color (not true color, but color) NV. In low light, the colors are pretty good. When it gets really dark and you need IR illumination, the color spectrum shifts because of the IR light, but there are still colors. The other cool thing is that the Sionyx will also record audio and video. The down side is that unless you zoom, the coyotes will look pretty tiny at 50 yards on video. You could zoom, but if you zoom more than a little, you will likely want to be stationary because walking around with zoomed optics, at night, is a disorientating challenge. The Sionyx will zoom quite a bit and still have a good picture, FYI.
Or, you could go with a helmet mounted PVS-14 and go with a larger size IR holographic/red dot sight, IR illuminator, and maybe an option IR laser.
I do believe that Pulsar and Rix both make DNV weapon sights that will handle that sort of recoil. The Rix will be considerably less expensive. For $600, the Tourer T20 comes with a functional IR illuminator that will get you out a couple hundred yards in less than ideal weather. I have used it and it is nothing amazing, just nicely functional. Full disclosure, I was part of Rix's pro staff program for that last several months, so have experience with their stuff, but am not affiliated with them now. The Sightmark Wraith Mini series that I would have suggested, is only rated for .308. Thermally, both Puslar and Rix should handle the job and Rix has a nifty Storm series that is $1100 for the S2 256 resolution (would not suggest because of lack of resolution) and $1800 for the S3 that is 384 resolution. The S3 is very capable with coyotes. I have used it at 240 yards and I have seen vids where folks made shots much farther, though it was hard to tell the coyotes were coyotes based on the anatomy you could discern.
Bering Optics also makes thermal weapon sights that should handle the recoil. My hunting partner actually did some of the field testing to substantiate this, but via rifles with .300 Win Mag and the like. If you went with Bering, I would double check with them, first.
There may be others out there, but off the top of my head, nothing is coming to mind. For example, there are some Brits that have used Hikmicro thermals on shotguns. Hikmicro is sort of new in the US and I know virtually nothing about them. Pard is a good budget DNV company that is now doing thermal, but I don't know the specs in their gear.
Also note that with things like electro-optics, the more you pound them, the more likely they are to fail. The companies are betting that their gear will hold up for the 3, 4, or 5 year warranties under normal use within the specified recoil rating. If you mount their gear on a .22 and shoot 10,000 rounds, the gear likely won't be any worse for the wear. 5,000 rounds of 12 ga may kill it. Or maybe it is 10k rounds. Who knows? Most people won't shoot 1000 rounds of shotgun per year at night (using DNV or thermal) on average so the company is safe warranty-wise. However, the logic here is that your off road truck may go 500K on nice smooth roads, no problem, but may have stuff falling of after 200k of driving the boulder strewn fields of some of the the desert regions of the US. Hard use does take a toll.
