If you wife's hair tool is truly 135W, no problem....you have a 2000W inverter. If it's some sort of heater/curler read the label closely.....it may be much more than 135W.
One other thing....if you put a large electrical load say close to 2000 watts on the inverter, a single 200AH battery may struggle to supply 12V to the inverter....something like a 1500W microwave or a coffeemaker or a blow dryer. You should be fine with loads of less than 1000W or so.
Of course this also depends on the lengths of battery cable runs and how large the cables are.
FWIW I have 350AH of house battery and a 1000W inverter. When I run my 700W microwave, the inverter pulls the voltage below 12V sometimes depending on the battery's state of charge.....these dips into the 11 volt range can trip some inverters low voltage cutoff/alarms.
2008 E350 RB passenger 4WD SMB penthouse
2013 KTM 350 EXC
2008 KTM 250 XCF-W
2000 KTM 200 EXC
2003 Honda Element