I have no experience with your proposition, but I can tell you that your heated mattress will require roughly 6 amps of DC power...far less than your TV's charging system should be capable of. Your 12v outlet should be fine for this. However, at full output (180w) your inverter will require roughly 14 amps of DC power...more than I would be comfortable with long term via a 12v outlet. You may consider hard wiring the inverter?
'04 GMC Sierra 2500HD, CCSB, D/A, MaxBrake, B&W Companion, Retrax Pro, AR Power Steps
2013 Rockwood 8289WS w/Diamond pkg
1969 Camaro RS/SS Convertible
Dad's old 1968 Empi Imp (recently recovered from under 21 years worth of boxes/dust)