Lead/Acid battery voltage changes with the amount of load. All those different voltages mean nothing unless they are all done when the battery is loaded the same.
Next, loose connections, connection corrosion, and long runs of thin gauge wire can cause low voltage.
If you are going to wire in a 400 watt inverter, why not use 10 gage wires to the battery with a 30 amp fuse. That way you can get 400 watts when you need it.
Static battery voltage at 12.8 is a good start. Static means no load and no charging for a few hours.
Next connect the 12 volt house system. Measure voltage again at the battery terminals. Measure the load end of a circuit. Do not change the load between measurements.
If voltage at the battery is significantly down after the load is connected, the battery is not capable of supporting the system. It could be a bad battery. It could be too much current being drawn from the battery. 5 amps from a 75 amp hour deep draw battery is a substantial load. 30 amps is extremely heavy, possibly too much. Voltage at the battery terminals will drop quickly as the load current increases. Measuring the current draw is the best way to load test a battery.
Voltage at the load end of a circuit being lower than the voltage at the battery terminals means the wire and connections between the battery and the load may be insufficient.
Use these two scenarios to isolate your issues.
Getting the battery load tested is also a good way to determine if the battery is bad. However, many automotive battery shops are set up to test engine start batteries. Deep draw batteries are often tested using a 20 hour discharge rate for standard purposes. (75 amp hr / 20 = 3.4 amps for 20 hours).
Lead acid batteries can be safely stored in freezing conditions. All lead acid batteries should be stored fully charged. Full charge takes 14 to 18 hours regardless of starting state of charge. A fully charged battery is unlikely to freeze anywhere you would choose to store it. Not storing fully charged can lead to freezing and sulfation. (Sulfation is when plates accumulate large crystals of lead sulfate.)
I wish you good luck and happy trails ahead!