You'd definitely have to skimp on the air conditioner to make this work, and spend a whole lot of money on solar panels.
A 42-ft motorhome is usually 101" (8.42ft) wide, giving a roof area of 353.5 sq ft, or 32.84 sq meters. Let's stipulate that the roof extends up on one side by a 15º angle, which increases the roof area to 32.84 / cos(15º) = 34 square meters.
In my part of Alabama, we get 3 - 4 kWH per square meter per day in June. Let's say it's 4. The motorhome solar panel has 4 x 34 = 136 kwH per day of sun falling on it.
The highest efficiency solar panels I've found on the Internet, from Panasonic and Sunpower, claim 22.5% efficiency. Let's say the marketing hype is true. Then you can get 136 * 0.225 = 30.6 kWh per DAY. That doesn't include battery charging and discharge losses, which probably will take you down to about 80% of that, or about 24.5 kwH per day.
Ramsond says their 15000 BTU RV air conditioner requires 3.5 kW just to start up (say it takes 1 min to start up, so we've used 58 wH), and 1.5 kH to run. You can run the air conditioner, and only the air conditioner, for about 16 hours off the stored solar power.
It's going to be hot for the other 8 hours while you charge the batteries again.
This doesn't look very practical, and I've just calculated the A/C load.
Modern RV appliances and appliances optimized for solar battery banks have a “soft start” in order to minimize the startup draw, so your numbers are off in that regard.
Go to the link to Precision RV that I provided and look at the several large installations shown that power the entire house functions of the RV. These retrofits are not cheap and they would not be selling if they do not function as advertised, RV’ers are not generally pie-in-the-sky leftie dreamers, they’re a practical lot.
I’ve been following this as it’s been developing for over two years now. Two years ago, yes, you’d have been correct. Running the A/C on solar-powered battery bank would have worked for at best a few hours, not overnight. Not so now.
In addition to “soft start” appliances, the inverter(s) play a major role as does choice of type of electrical appliance or A/C unit. Here’s an ideal Danfoss compressor refrigerator for such applications (there are larger models):
http://www.backwoodssolar.com/novakool-dc-refrigerator-model-r5810
In addition, mobile Danhard A/C units with variable speed compressors are ideal for such applications. The startup power draw is minimized as is the running power draw.
The same is true for True Induction cooktops, microwave/convection ovens, just about anything you’d put into an RV. The domestic, US RV industry has been rather slow to adapt to this, but it’s been innovated largely by the Australians who have taken the matter even further along, with ducted mini-split RV air conditioning systems, Danfoss compressor refrigerators with “eutectic plate” in order to utilize power from the engine alternator to get a jump on cooling the refrigerator while being driven. Large secondary alternators are also being used to maintain or increase charge to the lithium ion battery bank. Maybe it sounds like “cheating” to start with a fully charged battery bank, maybe that’s why your numbers indicate that the system cannot work. All I know is, it does work, I’ve been pretty thoroughly researching the matter for years as I mentioned.
Here’s a brand-new offering from Winnebago called the “Revel” however, that does demonstrate that the domestic RV industry is beginning to wake up to the new reality, a very handsome 4x4 Sprinter conversion with many of the appliances I discussed, with the exception of the A/C unit but they do use one of the most efficient rooftop units available (cost considerations I’m sure, since the very similar Sportmobile uses a proper Danhard unit, but they’re more expensive):
https://winnebagoind.com/products/class-b/2018/revel/overview