I think you may be approaching this backwards...and making things too complex.
Many inverters come with built in transfer switches...which means that the inverter connects to the shore power converter panel and it connects to the battery.
When shore power (e.g. the generator) is available, the inverter takes its input power from the 120 volt supply and simply passes shore power through the inverter's output circuits. When 120 volt power disappears, the inverter draws from the battery bank.
Since a 1000 watt inverter is NOT going to power your whole coach, and since 4 x 6 volt GC-2s won't power your whole coach for very long, there's no reason to try to kluge up some system in which you're using the inverter to run other 120 volt loads in your coach. Locate the small inverter somewhere conveninent to your modem/router and TV...and where it's convenient to feed it with a 120 volt line from your panel and 12 volts from your battery bank. Oversize the wire.
1000 watts = 84 amps at 12 volts.
You'd probably be wise to go with 2 AWG wire for your connection to the battery if you are running any distance at all. For a small margin of error, use 100 amps to calculate wire gauge. This will minimize losses in the connecting wire. This will be the most challenging part of the installation unless you keep the iverter very close to the batteries and distribute power from the inverter using a 120 volt dedicated circuit. In that case, 14/2 household wire (what's all through your coach for 120 volt circuits) can carry the power from the inverter to whatever location suits you. 14/2 can handle far more than 1000 watts...about 9 amps at 120 volts.
Your inverter will have at least two outlets...for internet and TV, but you can use a surge protector to add more outlets...say you need one for a modem and one for a router...plus the TV. You could also feed two 120 volt circuits so long as you don't forget the ultimate source is only 1000 watts and 4 GC-2s.
If you choose to run wire underneath the coach...the easiest path...use conduit to protect your wire from rodents. PVC or steel will work. Rodents like to chew insulation. Conduit is cheap and easy insurance to protect your wire, and it simplifies installation.
This would make a safe, semi-permanent installation with none of the shenanigans you appear to describe in your OP (frankly, I didn't fully understand the plan).
Best of luck...and as others said, there are many 12 volt devices out there, including TVs. Eliminating the inefficiencies of the inverter would make sense. Apply your inverter and wire budget to 12 volt devices.
__________________
Jim & Renee
2020 Jayco Jay Feather X-213
previously 2014 Forest River/Rockwood HW 277
2006 Ram 1500 4WD Crew with Firestone Airbags
Every weekend boondocking in the National Forests or at
Lake Vallecito.