32" tv , approx 100w, thats 8amps
8a * 12hrs = 96 amp hrs
or about 42 percent use of a 220 amp hr bank
thats at max power full bright sound up etc
all likely hood, power use would be even less
and NO you don't need to tear out walls, you might need to drill one or two holes
depends on where the inverter goes, and whether the wire goes inside against the wall, covered by wire molding
or under the RV, thru the frame and then up inside
its some wire, NOT a 'remodel'
that 12v outlet in the cabinet is only good for 7 or 8 amps max
might work, might work might not
depends on the wire feeding it and how much voltage drop is there when you turn on the inverter and TV
the only way to find out is to try it