I think you've found the problem. Your 12v wiring is too small.
If I understand correctly you want a long 12v cord on the inverter so you can use it in the LR or the BR while still plugged into your RV's 1 12v outlet. This is the wrong approach. You would want to leave the inverter near the 12v outlet and use a long cord for the 120v output.
If you are just needing a heavier cord on the inverter one could be made easily but I doubt anything is available ready made. There really isn't much available as far as 12v extension cords and such.
I think your best option would be to hard wire the inverter near the batteries. You could either use an extension cord to the TV or hook up the RV's appropriate circuits to the 120v output.