Modern RV's can leave the charger plugged in all the time, and it will not overcharge the battery.
Older RV's such as from the 80's or 70's had "Converters" that where not electronically controlled output, and it could be charging at anything from 12.0 to 15.5 volts - much dependent on the input voltage. You do not have that type of converter.
If your charged battery voltage is between 12.8 and 13.8 volts, you will not boil off an excess amount of water, and the voltage is fine for long term leaving it plugged in. Even if on the higher side of 13.5, just add some water every 2 months, and it will be fine.
Allowing the CO meter to discharge the battery to far while not plugged in is a sure way to freeze it, and that will destroy the battery. Leaving it plugged in will not damage anything. If you leave the refrigerator on electric, it can cost you upwards of $35 a month, but just the battery charger working will be insignificant $2 or so on your electric bill.
Good luck,
Fred.