I would not hesitate to travel 117 miles off road on a F tank, if you have a gas station at the end. I routinely do that here on backroad trips albeit it is not as remote as ID. Carrying fuel in the car is dangerous for fumes and in the event of a crash/rollover. I would recommend bringing extra food/water/gear etc in case something crazy were to happen so you could wait a couple of days for help or walk out. It looks like that road gets a fair amount of summer traffic.
EDIT: If you are concerned about range, perhaps strap a metal gas can on one of these:
Amazing deals on this 300Lb Capacity Atv Cargo Carrier at Harbor Freight. Quality tools & low prices.
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It get's it outside of the car and removes the fumes/fire issue. I have this rack and it's high enough that the GX exhaust is routed underneath it (but I'd still put the gas can on the other side away from the exhaust). I cannot comment on this specific trail enough to know if you'd routinely wack the carrier on rocks/etc.