To turn on the car once you're inside, can I suggest holding the fob in the position by the cigarette lighter? Note that nothing will change on the display, but you can press the brake and the car will turn on. There's a picture in the manual of where precisely to place the fob. In my manual it's page 17 under "Unlocking When the Key Doesn't Work":
"To drive Model X after gaining access to the cabin, place the bottom of the key against the center console, immediately below the 12V power outlet, then press and hold the brake pedal to start Model X."
I had the same scenario in a state park in Texas with no cell service with m Model S (but I'd expect same behavior from Model X). I used the fob to open the door, but when I pressed the brake the car indicated the fob wasn't found. I tried cell app to no avail, I tried rebooting, I tried walking away for a while, but no luck. After an hour I finally remembered that trick, but still took time to realize I needed to press the brake even with the "fob not found" message on the dash. The next day, everything was normal, but I did replace the battery just in case. I don't remember what software version this was, but it was maybe 2 months ago.
I'm still unsure what triggered this misbehavior. My oddball theory is that since I had squeezed the fob in a waterproof case with my cell, it likely continuously pressed the top of the fob for an hour. I've never done that elsewhere. Theory falls apart at this point though. No idea why it could still unlock the car but not be recognized to start it. A second kayaking trip to a DIFFERENT park, but using the same waterproof case, resulted in the same behavior, but I was much faster starting the car that time.