I have a 14 sport with pushbutton start and touchless doors., so fob just stays in the pocket.
2 times it has locked me out, no unlock, no lights coming on, ie everything dead,
Last time it happened I was up town at night, and I did not have the door key as I had left the key ring out at home, So I bought new battery for the fob, no help.
Phoned wife, in bed ,So you know how that went, almost better to walk home.
She brought up her fob, still would not work, but I could unlock the door and use the fob to push the start button and the truck started.
Make sure to keep the door key with you , so you can open the door.
This happened one other time, and it would start with the fob pushing the button, as it tells you in the book.
After wards it always has worked fine, last time was about 3 yrs ago , and the time before was afew yrs as well.
May not help you , but I don't think it is a fob problem if some stuff works. a fob works or it don't, if you have pushbutton start and lights come on then you have a signal.
Have you put a meter on the battery and seen what battery volt is while you try to start?
It should be close to 12v, thou I have never check on this truck to see what crank voltage is.
I still have the factory battery in this , it sits in the garage , about 82,000KM's now as O have been retired for 14 ys, and don't pull trailer around.
Happy New year to you all , sorry about the long post