If you have a 6kW battery backup, then barring huge power demands, you should be able to keep the whole house running. the system should seamlessly switch to battery feed if there is a mains drop out. If the system does not do that, look at systems that do. The obvious one is Tesla. Tesla Powerwall compared to other of similar sized system is not expensive. They just use a bigger battery than most
Powerwall Specs
https://www.tesla.com/en_gb/powerwall
- Energy Capacity
13.5 kWh
100% depth of discharge
90% round trip efficiency
- Power
7kW peak / 5kW continuous
Quick backup transition
Pure sine wave output
- Size and Weight
L x W x D
45.3" x 29.6" x 5.75"
1150 mm x 753 mm x 147 mm
251.3 lbs / 114 kg
- Installation
Floor or wall mounted
Indoor or outdoor
Up to 10 Powerwalls---------------------This feature for power hungry properties
-4°F to 122°F / -20°C to 50°C
Water and dust resistance to IP67
- Certifications
Meets local safety standards and regulations
- Warranty
10 years