The three 250w panels will run the 5000 BTU A/C mid day when the sun is shining and that's when I need it. I don't need it when it clouds up, it's cool at elevation in the evenings and we run the furnace at night. The system is designed to meet our needs which are very specific. It took over a year to work it out, find the right A/C, panels, controller and inverter that could handle the start up surge.
That said, even if you had the batteries to run it over night, three panels couldn't support it. There is only about 10a left over while running the A/C and that's not going to replace what you used over night. Also considering there will be evening times when the panels are not putting out full potential, the batteries are already running it partially, not just when it's dark.
I did see a you tube where like me the guy could run the A/C while the solar was going but at night the inverter shut off even though his batteries were full. Without the solar the start up surge was too much for his batteries even though the controller display and a Kill a watt couldn't see the voltage drop. It happens so fast that even my clamp on only has a chance to jump about 7a before dropping unless the compressor struggles a bit, then you see it jump to over 120a. Run it for a few hours draining the bank farther and the restart after the compressor cycles will be even harder. Two 6v wont start it and I'm willing to bet four 6 volt wont be able to do it after running it for a while.
Remember I'm talking a efficient 5000 BTU unit, not the roof top. Just getting one of those to start is impressive, restarting after running it for hours....
Panel prices have been low but I think the Chinese pushing the cost down is a response to the tariffs. I have heard more than once that prices are going up due to the tariffs and to get your shopping done now. One of the people was the owner of the company that made my panels and our deal was done, so he wasn't saying it to sell me anything. My 750w was $525 and his 280w panels were $220 but too big for me to use.