Given your budget and additional constraints (new, amazon prime, etc) it would be useful to re-confirm your wife's intended riding to figure out the best value for your money. I totally understand how living in rural Alaska would make this difficult.

how often do you think she'd be making 50+ mile trips? - Depending on terrain and her load that's probably 4-5 hours or saddle time. It's completely do-able but if this is something she wants to do regularly it would be worth your time and money to reach beyond amazon and find something that's going to fit well and be comfortable for that long of a ride.

what's the biggest town within an hour drive? what's the biggest town/city you go to a couple of times throughout the year? - Just to double-check what options you may have for working with a local bike shop. A 30 minute visit is probably all you need to get her sized up on a bike that fits well; they can ship to you if there's nothing on the floor that works.

is riding on the snow/ice something you want to have as an option? 10 years ago we wouldn't even be having this conversation but today you can buy capable bikes that work year-round at Wal-Mart for well under your budget.

If you just want to roll the dice on a cheap department store road bike you can't go wrong with the GMC Denali (http://www.amazon.com/GMC-Denali-Black-22-5-Inch-Medium/dp/B00FNVBS5C/ref=sr_1_2?ie=UTF8&qid=1456753692&sr=8-2&keywords=touring+bike). Don't even bother reading the reviews - this bike is worthy of the $200 asking price. I have $4,000 invested across two bicycles right now and would buy one of these in a minute.

For a snow-worthy bike do something like this Mongoose that you can buy at Walmart as well. http://www.amazon.com/Mongoose-Mens-Malus-Tire-Silver/dp/B00TYB8Q7A/ref=sr_1_23?s=outdoor-recreation&ie=UTF8&qid=1456754198&sr=1-23&refinements=p_72%3A1248957011

There are also plenty of Schwinn hybrid options under $200 that would be fine for around-town use.

Finally, either buy a bike that costs less than $300 or more than $900. That's where the value is. Between $300 and $900 you're either paying a premium for a brand name bike or "saving" a few hundred dollars on a bike that's heavier and less comfortable to ride than you really want.