The key facts about your A/C are
Power Requirements 1/230V/50Hz
Cooling Operating Current 7 A
7 A x 230 V = 1610 Watts. UPS specs are in VA not W due to power-factor corrections which I'll ignore:
Your 800 VA UPS is probably not a good choice for a 1610 W load.
I was headed this way until about 2009. In the most modern and expensive setups you might have all the devices communicating and coordinating. In most, each one has its charging parameters that it's trying to meet, and the only communication is the voltage on the bank.
With multiple, uncoordinated charge controllers you may exceed the optimum charging current, and you should expect to have disagreements among uncoordinated charge controllers (and a water turbine with no charge controller) as their charging set-points and algorithms are almost certainly different.
If you have the money to put into it, or you see it as saving money on the battery bank's lifetime, a set of networked, coordinated smart charge controllers would certainly work more optimally (and can all work at the same time) - but you should also realize that any system that won't leave you in the dark for large parts of the year is going to have some excess capacity when everything is going well, charging-wise.
You can't simply add more battery without making it really hard to get it fully charged and equalized, and you can't reduce power sources without finding yourself in the dark after a few dry, cloudy, windless days. So sometimes, you are going to be dumping excess power. If you can do something useful with that power, great; (laundry is a good one, often) otherwise it's just part of the cost of having power more of the time. In the case of a fuel burning generator and a wind turbine, common sense would be to shut down the fuel-burner on a windy day.
Best Answer
Since you keep a 6 amp charger on the battery you may have boiled the water out of the battery it shows a full charge until a load is put on the battery then the voltage drops because there is not enough liquid to maintain the voltage/ current demand. If you can open the battery and add distilled water and this may provide some run time. Next find a quality Float charger, these do not over charge the battery like a trickle charger will and boil off the water. Last you might want to look at a larger battery, placing batteries in parallel will increase the time the system will run but if one battery is bad or has low internal resistance it will discharge the other battery unless there is an isolator. A larger deep cycle battery is what you need standard car batteries are not designed for deep cycle. Get as large deep cycle battery as you can afford (higher amp hours) golf cart batterys are also a good way to go many of these are 6v so it would take 2 in series.