You are correct Pat. I just read through the owners manual for the Dometic CF35, and here is what it says:
Using the battery monitor
The device is equipped with a multi-level battery monitor that
protects your vehicle battery against excessive discharging when
the device is connected to the on-board 12/24 V supply.
If the cooler is operated when the vehicle ignition is switched off,
the cooler switches off automatically as soon as the supply voltage
falls below a set level. The cooler will switch back on once the
battery has been recharged to the restart voltage level.
a
Caution – Danger of damage!
When switched off by the battery monitor, the battery will no longer
be fully charged. Avoid starting repeatedly or operating current
consumers without longer charging phases. Ensure that the battery
is recharged.
This is how to select the battery monitor mode:
CF-25, CF-35, CF-40, CF-50, CF-60
➤ Switch on the cooler.
➤ Press the "SET" button (Fig. 4 4, page 4) three times.
➤ Use the "UP +" (Fig. 4 6, page 4) and "DOWN -" (Fig. 4 7,
page 4) buttons to select the battery monitor mode.
✓ The selected mode then appears in the display for a few sec-
onds. The display flashes several times before it returns to the
current temperature.
I
Note
When the cooler is supplied by the starter battery, select the battery
monitor mode "HIGH". If the cooler is connected to a supply battery,
the battery monitor mode "LOW" will suffice.
If you wish to operate the cooler from the AC mains, set the battery
monitor to the "LOW" position.
There was a table on what the voltages are for the different modes, but I couldn't get it to print. If you are interested in looking at it, or any of the rest of the manual, here is a link to the online version:
Using The Battery Monitor - Dometic CoolFreeze CF-35 Instruction Manual [Page 17]
So, as long as that is the case, your original idea of the booster as some added insurance may not be a bad one, as long as the Dometic does it's job of shutting down before discharging to the point where damage to the TV battery is done. Sounds like you need to make sure you charge it long enough after a low voltage shut down to make sure it gets fully recharged in order to prevent battery damage.
Good luck, and let us know how your choice works out!