I suffer from sleep apnea and have to stay connected to AC power at night. I have a 300W panel, MECER 2000VA inverter with MMPT controller, and 2/12v 100ah batteries. When the inverter is connected to AC, there is no interchanging power tracking - only power tracking between solar and battery...
This boat has sailed, but I had similar ideas, and hooked my RCT 2000VA UPS to 2/12V 18a batteries connected to 300w solar panel. I use the solar panel to load the batteries and run my security cameras, Wifi, TV, 2 computers without ESKOM during day time and during load shedding.