Depending on how big an aircon unit you want to run and how long you need to run it, the trenching might be cheaper. Aircon, stoves, ovens, and well pumps are all destroyers of batteries because they have such high constant draw.
Just for some rough numbers, a 12k BTU inverter style mini-split, which are about as efficient and low power as they come, are about 1200w at full bore and drop to somewhere around 800w to just keep things cool. Assuming you go with a 24v system (which would probably fit your needs best) that's 800wh / 24v = 34a/hr or a little over 1/3 of a 24v 100Ah LFP battery each hour you want to run it. Just for arguments sake lets say you get one of the 200Ah 24v rack-mount batteries, that's 6hrs maybe per day of aircon, not including anything else you want to power like lights or laptop or TV or noodle maker. Rule of thumb is 3 days battery bank, so you're looking at $4500 just in batteries for the aircon.
Then you're going to need panels and SCC to refill that, and if you do some serious looking and go with used panels you'll pay about $0.50-ish per watt of panel, and to recharge hose panels with the average 4-sun hours per day, that's about 3Kw of panels to recharge your battery bank in 1 day, or about $1500-$2000 in panels.
Then you'll need the AIO you're thinking about which is about $700 for the Growatt 3Kw 24v model. That's not going to have enough amperage to fully charge your batteries in a day so you'll need at least another 60a MPPT controller to parallel up which is another
$300 -
$600 depending on how fancy you want to get.
You mention you're in a pretty cloudy area so you're probably going to need to significantly over panel your system, so double the size of the array to 6Kw for $4000 in panels and a LOT of space in the yard.
So, rough numbers here:
Panels - $4000
Batteries - $4500
SCC's & Inverters - $1000
Lots of yard space for racking
Storage space for batteries & parts
You can doodle the numbers to your needs to get a more accurate number, but you get the idea of what it's going to take. You'll want to do a full power audit of what you currently use then add in the Aircon system's numbers to get a more accurate estimate. I'm rounding high here a bit because it makes the napkin math easier. I'm sure once you start filling in the blanks on the power audit the numbers will be smaller.