Your tow vehicle should be acting as any other charging source in your tow vehicle.
How do you know that the source of power for your rig is from the TV and not the batt, when you have the batt installed?
The TV charging system is only responsible for maintaining a charge on the battery that activate the brakes in the event of a breakaway. It does this by acting as a charger, the same as your onboard charger/converter.
Your TV system is effectively parallel with all your other charging sources through the terminal ends at your battery, no single charging device is exclusively tasked to do a job, each charging source can act independently. Of course your charger/converter is not activated when driving, so your TV is the only source of power when traveling. In the case of my solar both systems are charging or providing power to the batts at the same time.
I can see no reason why power from your TV is not reaching the DC systems on your rig, unless the TV is not providing power to the rig, or there is a break in the TV circuit. Have you taken a multimeter and starting at your 7 pin checked for current? When you disconnect the battery could you be disconnecting the TV wires inadvertently from the terminal ends of your cables? ex. are there any member wires that would otherwise be connected to the battery post, dangling?
Could the TV have a safety mechanism to prevent you from discharging your TV batts in the absence of a breakaway/house batt? Do your signal lights work?
I would start at the 7 pin and check all the pins with a multimeter. Then I would move toward the batt looking for a break in the circuit.
IMHO if your DC system in your rig is not being provided with 12v DC by the TV, neither is your batt when installed. Meaning that your breakaway system is at risk of not being charged.