A/T tires on a Prius Prime?

A-T type tires won't give you any more sturdiness than any other good tire. I'd suggest Michelin Defenders, there are several variations.
All you really need to do is take a spare along on the trip.

 

I can tell you that 205/70R15 is indeed the biggest baddest tire size that works properly on a Prime. Short of adding a 30mm or 40mm lift kit at all 4 corners, using that size and pumping them up to max pressure given on the sidewall of the tire, is the most ground clearance you're going to get. Try not to overload the car with too much heavy stuff when you're needing ground clearance, the car has little enough clearance with just the driver in it.

In my 2017 Prime (year does not matter so far) I've driven 700 miles on the original Toyos, 63K miles on Michelin X-Ice3 205/65R15 XL 99T, and 39K miles on Michelin X-Ice3 205/70R15 96T. I took both sizes offroad to places you would not believe. The 65 series up a jeep trail above Arco, ID. The 70 series to an undisclosed location near a secret plane crash site in NV. Both are a challenge for something like a RAV4 or Highlander, but you can make it in a Prime with oversized tires, if you pump them all the way up and don't mind risking minor to moderate body damage if you misjudge how big a rock is, etc. I only bonked the bottom of the body behind a wheel well once, and it was my own fault for not getting out to check how high a big rock was. And I was probably 300 pounds over the car's GVWR at the time also, due to a full 50 liter fridge, 50 pounds of water bottles, 100 pounds of emergency water, hundreds of pounds of tools/lumber for another mission on that road trip, etc.

Personally I'd recommend a high-end winter tire with XL load rating, over an aggressive-tread all-terrain tire, though either is about as good as you can get for your mission. Continental's 205/70R15 XL 100T or the General Altimax Arctic 12 with those same size, load index, and speed ratings, would be what I'd choose for your mission. That load index of 100 is pretty serious. XL (extra load) tires don't reach their max carrying capacity until 42psi, unlike SL (standard load) tires which reach it at 36psi. But in either case, I'd pump them to the max pressure given on the sidewall, to raise ground clearance. The T speed rating is 118mph, better than Prime stock.

The only stronger tires I know of in that "Prime max" size, are some LT "light truck" or van tires, for instance:
Goodyear Durasport 205/70R15 with load index 106 (/104 if used on dual rear wheels)
They are a huge bargain if you are looking for sidewall strength and safety margin on load capacity.
They are considered "6-ply" rated, Load Range C, rather than 4 for most passenger tires.
The ply rating is not a count of actual plies, just an equivalent strength rating.
Unlike some similar truck tires, they still have the S speed rating for 112mph, same as the stock Prime tires.

 

More specifically, the Continental VikingContact 7 model, in that size/spec seems to be great, fully comparable to the Michelin X-Ice3, but with load index 100 instead of 96. Generally $600+/set, but findable online for $475ish/set, minus $70 mail-in rebate for prepaid Visa if you jump though all the hoops proving purchase/installation without making a mistake in documenting it.

 

Highly recommend the Michelin Defenders!

I just recently replaced the factory TOYO tire with the Michelin Defenders and road noise is night and day.

Also, Costco is currently having a $150 off Michelin tires.
Set of 4 Michelin Defenders for: $464.88 (w/ $150 off)

https://tires.costco.com

 

I saw a youtube video talking about how your "gas" mileage would decrease if you bought sturdy tires.

Is that wrong?