Similar problem - replaced the battery - problem did not go away. I replaced the battery in the remote and it went away for over 6 months. Now I am getting the error again. Ford dealer did not have a clue and I do not know why the remote would have and effect but it did.

 

How much do you drive the truck..? Have a friend who puts very few miles a month on the truck and she gets the same warning message..

 

2013 F150 XLT-5.0 V-8, 14,978 miles. I live in the South, so no cold weather. Out of nowhere my truck started this low battery thing. Radio would not stay on, lights telling me the battery was low. Since the battery was almost three years old I went to Wal-mart and got a new 65 size battery with 5 yrs replacement for about $110. Installed the battery and had the same problem. Checked alt with meter and the charging system was good.
Took truck to Ford Dealer, they said this was new to them, never seen this problem before. Sounds like BS to me. Ford has a repair manual with tech notes, the one they used was SSM 44750 2013-2014 F-150 Battery Saver Mode or System Off to Save Battery Message- Load Shed Strategy Diagnosis and Repair. What they did was reprogram the BCM, this requires the use of a IDS hooked up to the internet, so its not something I could do myself.
Really all they did was re-boot the onboard computer, no parts were replaced. My truck was under warranty so this did not cost me anything.
This fixed the problem, for now, not to say it won't come back later.

 

I had a 2011, 2013 and now a 16 that does it. Ford techs couldn't fix any of them. Finally found the problem. Ford programs the alternator to shut off to save on fuel . Take lots of short trips and it doesn't recharge enough. Torrie at Unleashed Tuning has a fix that he can put in his custom tunes. i am sure other tuners could too

 

Mine did that -- a new AGM battery and no problems since. Plus, the radio plays for quite a while before it shuts down.

I also watch my battery voltage (on a separate gauge) to make sure it's charging properly. I do see the BMS cycling the alternator on longer trips.