Congrats and good luck! In helping a buddy with a sheep tag in ID we came across more goats than sheep on our first trip in. Has made me consider throwing my name in the hat. Maybe someday...
Magnification is not as important, IMO, than other attributes of the scope for shooting long range. A 9x at 800 yards makes it appear less than 100yds away...
In that price range I'd be looking at these and in this order:
https://www.eurooptic.com/C557-NF-Nightforce-SHV-4-14x50-F1-1-MIL--illuminated-Mil-R-r.aspx
https://www.gaprecision.net/ga-store/optics/bushnell-lrhs-2-4-5-18-x-44-g2h-reticle.html
'Twer I in your shoes, I'd have the Tikka barrel shortened and threaded before buying another rifle just for a shorter, threaded barrel.
For those concerned about length, be sure to look at suppressors of the reflex design. Some of the length is back over the barrel. Like this for example...
Might be worth the time to look up the posts Eli Grimmet (sp?) made on the subject of drought and pronghorn horn growth. The short version is that they feel that the conditions the year the buck is born is more important the conditions of the year of harvest for influencing horn size.
I'm a BIG fan of fixed power scopes for younger/newer shooters. If used is okay, we've had good luck with the Weaver K6 on a rifle for my two boys. If new, I'd be looking hard at a Leupold 6X; either the 36 or the 42 though I recognize they are over your budget.
Of the two you listed, I'd...
An Affinity is at the top of my list if I opt for an inertia semi-auto. That said, I will likely NEVER shoot a 3.5" shell so would opt for the 3" version. I can't imagine you wouldn't be happy with one.
I've not pulled the trigger on a sheep, but I'd say I've hunted them. I helped a buddy fill two sheep tags in Idaho. One in the Frank Church Wilderness (took 2 trips) and one in Bruneau Canyon along the ID/NV border.
In a previous job I've helped change grazing permits to allow for the...