This sort of question gets asked here quite often, and two of the major mistakes people make are overestimating the output from their solar panels and overestimating the usable power from their batteries. The good news is that you've been quite realistic with the first one.
If your calculations and measurements on power requirements are correct, you're looking at using 15kWh a day in total, which makes an average consumption of 625W. From the sound of it, this is pretty much constant, which isn't alway the case and would need to be factored into the calculations if not.
Now, you need to think of your system as operating in two distinct modes:
- 17 hours without sunlight, where the 625W load has to come from the batteries via the inverter
- 5 hours of sunlight, where the solar panels have to provide the 625W via the inverter AS WELL AS recharging the batteries, because you can't use the batteries to power the load while you're charging them. And that's the third mistake people regularly make with these sort of battery backup designs
Next, you need to add in the power lost in your inverter: let's go with 90% efficiency, which means your average load is now 694W (or 700W for a nice round number). As per mkeith's comment, although a 2kW inverter seems plenty, you need to be sure it can supply the current needed to start up and run your a/c, the former can be twice the latter or more.
In the first case, 700W multiplied by 17 hours gives you 11.9kWh, which is the usable capacity you need from your batteries. This is not the same as the rated capacity, because you really don't want to be charging them and discharging them completely (in fact you'll never get anywhere near the full rated capacity from them). There's a lot of debate over exactly how fully you should charge them and discharge them, but it's certain that going up to 100% and down to zero will stress them far more and reduce their life drastically, especially in a hot climate such as yours. Tied in with this is the correct way to charge them, which is using a CC/CV algorithm that gets them up to about 80% quickly but takes a lot longer to get to 100% (which is why electric car manufacturers often specify the charge time to 80% as it's a lot more impressive that the time to 100%). You haven't mentioned using a proper charger but that's the fourth mistake commonly made - a BMS is not a charger, just a protection mechanism to stop things going dangerously wrong.
Taking a more realistic estimate, aim to charge to 80% and discharge to 20%, which means you're getting 60% of the rated capacity. That means your rated battery capacity needs to be somewhere around 19.833kWh, so, pretty much, 20kWh.
Now comes the second stage. You have 5 hours of sunlight to get that 12kWh (approx) of charge into your batteries, which works out at an average of 2.4kW. It might be wise to allow for maybe 4 or 4.5 hours to get the 12kWh, meaning you have time to get a little beyond the 80% charge at the slower CV rate. Let's go with 4.5 hours, which means charging at a rate of 2.67kW. Now you need to add on the 700W going to your load via the inverter while the batteries are charging, which makes 3.37kW in total. Finally, allow a bit more for losses in the MPPT/solar charge controller, if we call that 90% as well you end up needing 3.74kW from your solar panels, which is a bit of an increase on what you were originally thinking.
Obviously all the figures can be tweaked to some degree depending on your degree or optimism or pessimism, but they won't be far wrong. And as you note, good insulation can make a huge difference to the power requirements for air conditioning, I've seen loads of buildings with aircon running flat out because they have a metal roof and a suspended ceiling with nothing in between to stop the whole thing turning into a giant radiator.
