GFCI vs. 40v 5ah battery for water safety
It doesn't need to be a GFCI outlet.  It needs to be GFCI protected.  GFCI protection is conferred by having any particular outlet obtain power power from the LOAD side of a GFCI device somewhere.  On most string-topology circuits, a single well-placed GFCI device will protect the whole circuit. 

If you stick a GFCI tester in there, push the button and the outlet goes dead, you're sympatico.  

The bigger problem is the temptation to bring cords to the pool area

Your #1 safety risk is the fact that people wanting to plug things in, in the pool area, feel a need to run an extension cord from some distant location. This requires a social-engineering solution.  You need to rearrange your home to suppress that desire.

So either put GFCI protection on every outlet they might ever dream to tap... or, install a plethora of outlets, actually poolside, that are protected by a GFCI device somewhere.  I prefer to have my GFCI devices indoors, so I make extensive use of the LOAD feature of GFCIs. Now there's no reason for someone to run an extension cord to the pool area; there are outlets all over the place. 

GFCI is safe as you're going to get

A properly installed GFCI setup, with the protection device some distance from the outlet, is as safe as things can be.  It will trip the GFCI if hot or neutral current takes a path other than the normal one.  

Higher DC voltages bite.  HARD.

Electronics and car people tend to believe DC is docile and harmless.  It is at 12V, yes.  However, somewhere in the 20s it starts getting a personality, and by about 60 volts it goes full honey-badger.  Can 40V bite you?  Oh, yes it can! And it won't let go, either.  So ixnay on the 40V battery pack.  I encourage you to develop a little bit of fear of it. 

TVs are 12 volts, anyway

Besides, now that TV tubes are gone, lot of TVs these days actually use a low voltage, typically 12 volts.  Better off just using a battery pack that size than do a double conversion.