The short answer to your question is YES,
But there is a much longer answer:
Inverters come in 2x2xMANY types and sizes
First split: True sine wave v/s MSW (Modified sine/square wave) Go for TSW, I used to think the MSW units were more efficient and thus would last longer on a given battery charge.... Till I did the research and found they are very nearly identical in effiency, might make 10 minutes difference on my batteyr bank, if that.
MSW has a list of "Things that may not work well" This includes all sorts of electroincs, Night Lights, electric blankets, anything with a digital timer, among other things.. Even one coffee pot I used to have.
TSW the list is much shorter: (Anything needing too much power).
The next split: Stand alone/Inline.
Inline units work like a UPS, they replace your converter as well There are advantages to both these intergrated converter/charger/inverters and to separate units, I can make no recommendation,, I have both an intergrated Inverter/charger (Prosine 2.0) and Converter (Intella-Power 9180 with wizard).
BOTH are top end converter/chargers.
The MANY is the size, 300 watt, 600, 1000, 1500, 2000, 5000 10000, you pick the size.
How much do you wish to power?
Modern TV's with support electronics can often get buy with a Samulex 300 Watt TSW, Add in a good size HOME THEATER and you may wish to upgrade to 600 watts, add a Microwave, couple computers and such, 1500-2000 watt,
Air conditioners need around 3000-4000 watt (each) but read on,, Residential fridges need a few hundred watts.
Batteries: General recommendation is at least one pair of GC-2 six volt batteries (The two together make about a 8D or 4D battery 220-230 amp hours at 12 volt, but at half the weight of the larger 12 volt are easier to wrangle, they are DEEP CYCLE (not MARINE/deep cycle) and thus you can take them down to half full safely if you re-charge promptly)
One pair per Kilowatt, so a 2,000 watt inverter needs 2 pair, a 5KW inverter needs 10 pair. That is over 1,000 pounds of batteries.
Each pair holds about 1KWH of usable power at the overnight rate.. Perhaps a bit less.. Your A/C burns more than on KW every our so again, overnight is 10 pair.
Hope this helps.
Most RV's are wired like this.
Shore/Generator--Main Breaker box---Inverter---Sub panel
Main box --> Inverter, Fridge, Water heater, Some outlets, Air conditioner(s) and cut the lead to the converter (or in my case unplug it, or not as you wish, depends on the convrter).
Sub panel--Televisions, radios, entertainment stuff, computers, Microwave and GFCI chain.
On many RV's the "Sub panel" is built into the inverter, Mine it is not.
Normal operation (I do not do it this way) If I have shore power the Inverter/Charger senses that, passes shore power (up to 30 amps) to the sub panel loads, and charges the batteries (Normally disabled in my RV), should shore power be lost, it instantly switches to inverter mode, sucking power out of the batteries so my dinner continues to heat, the TV does not even blink. Wen shore power is restored it instantly switches back.
(In my case I use a different charger, but this is a matter of choice, in truth I rate the 3-stage + converter in the Prosine I have equal to the 9180 I use, but,,, For other reaons (Matter of personal preference) i like the 9180.