Short answer: Not long enough! There a few things to factor in when asking how long something can run on an inverter…
First – Can the inverter run it at all? For that, we need to know how many watts an item uses. While some appliances publish this information, most do not or only publish the amperage. To convert Amps to Watts it's Amps x Voltage = Watts. A typical RV AC unit uses about 13 Amps. So, 13 Amps x 120 Volts = 1560 Watts. Also, consider a minimum of 10% power loss converting DC to AC (what the inverter does). So, that's about 1700 Watts. Considering that's not the only thing running at any given time, a 2000W Inverter will be cutting it close but a 3000W Inverter will do it just fine. Note: many inverters can also provide surge ability above their rated capacity for short periods of time, which can help with the initial surge from an AC start. The batteries also need to be able to handle this.
Second – Can your batteries supply enough DC Amps for your Inverter? For this, we use the same formula in reverse. Assuming a 3000W inverter, it's 3000Watts / 12Volts = 250Amps. So, the battery bank needs to be able to supply 250Amps Continuous to max out the inverter. Use the same formula for the AC: 1700Watts / 12Volts = 142Amps.
Third – How long can I run it? Batteries are typically rated for both Amps (how many Amps it can supply continuously) and Amp Hours (Ah) (how long it can provide those amps). Let's take a 100Amp / 100Ah Lithium battery as an example. Well, right off the top, we know it will not work since we need 142 Amps. So, scratch that, and let's consider THREE of them in parallel (what we have currently). Now, we're dealing with a battery bank capable of 300Ah. That is to say, our battery bank can supply 300 Amps for one hour, 100 Amps for 3 hours,50 Amps for 6 hours, and so on. If we take the 300Ah divided by the 142Amps, we get 2.11. With no other loads, we could run one AC for about 2 hours.
So, as you can see, without a HUGE and expensive battery bank, running AC's on Inverter isn't very practical. Adding a ton of solar to offset some of that would improve things, but it's still all about the batteries.