There are threads on this forum about what size dish to use in Alaska, for each of those two satellites.
Normally down here in the lower 48, your quad output, dual satellite LNB will do exactly as advertized.
There are two kinds: DishPro, and Legacy.
The DishPro will have a big black/silver dp logo on the front of it.
The one thing I don't know about, is determining how far apart the focus points would be, given a non-standard dish.
If you can't find any info on the subject (and I would suspect someone here already knows), then you could certainly find the focus points easily, with a signal strength meter, and hand-holding an LNB at the focus.
(I'm not blowing smoke here, as I've done similar for other purposes)
If you can get sufficient signal at the spacing of the two LNBs in your quad, then you're good to go.
If it turns out that your focus points are closer or farther apart, then two adjustable LNBs would be a better way to go...
... and you'd need an appropriate electronic switch, too.
So, wait for other posters with better info, and do search the forum for more info , using the keyword: Alaska
Oh, one more thing.
You'll have to either rotate that LNB or the entire dish (and it's not got a mount to allow that) to dial in the correct skew.
Another reason most people use two LNBs, each on a separate dish , I suspect.