Remove the AC cable from your power center going to the TV/ SAT and attach it to the AC output of the Inverter / Charger. Run a new AC cable from the old breaker to AC input on the I/C. Run DC power cables from the I/C to your existing batteries.
When you have generator or shore power the I/C sill charge the batteries in parallel with your current charger / converter, and pass power to the TV/SAT. When there is no AC power, the I/C will generate AC for the TV/SAT, pulling power off the batteries. Figure 1 amp hour of batteries consumed for every 10 watt hours used. There is also power wasted when the loads are off by the inverter's load sense. Add at least 1 battery to your current bank, and go for it. 100 watts or more of solar on the roof would be nice as well.