Can you post more about the battery? What exactly do you have?? Cold cranking amps are not needed for a tt, but for starting an engine. There are dual purpose marine batteries you can use in an rv, that list cca. If you have a car battery, it will not hold up.
Really I think you have an issue with you converter not charging the battery. If your plugged in the converter continually produces 12v power for the systems and to keep the battery charged.
You need a voltmeter to confirm the issue. Without a voltmeter, check the 120 volt circuit breakers. Maybe the one for the converter is tripped. You may also have a loose wire connection. You will need to pull the circuit panel, and check all the wire lugs. Make sure to do this with the shore power and battery disconnected. Loose wires happen frequently.
If you have a voltmeter. With the shore power connected. Check the voltage across the battery terminals. If the converter is working and the battery is charging, you will have a reading of 13.7 volts or greater. If you have a reading like this your converter is working and charging. If the voltage reading is under 13.5 volts, you may have a converter issue.
A fully charged battery that has rested for an hour or so after being removed from a charger should have a voltage reading on 12.7 volts.
Added thoughts.
How do you charge the battery back up at night? I reread your post. It takes crazy amount of hours to charge the tt battery if your using the 7 pin wire harness. Might be just getting enough power into the battery to make it through the night.
Have you verified the 12 volt fuses? There is one inline fuse at the battery, and two in the fuse box, one for reverse polarity protection, and another for load protection. These will be 30 or 40 amps
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