Battery charging

Posting the make / model of your converter would help. If it's a WFCO or Progressive Dynamics it won't hurt the battery by keeping it plugged in.

A fixed voltage charger is fine in cold to mild summer temps, but in hot weather will over charge batteries.
If you do much dry camping on a generator, you will want a 3 stage charger for faster, safer charging. And you can leave it plugged in all the time.

I assume you have a Magnetek/Parallax converter. These are OEM on most RVs. They are single stage chargers that apply 13.8v to the battery. 13.8v is a compromise, to low for fast charging, to high for maintaining a fully charged battery. They are notorious for boiling the water away in the battery. Keep a close watch on the water level in the battery.

For long term storage I use a Battery Minder. It keeps the battery charged with little maintenance required.

I also replaced my Parallax converter with a Progressive Dynamics 3 stage converter that charges the battery fast, then reduces the voltage for maintaining the charge.

Read The 12 Volt Side of Life. You may find it informative and helpful.

Never thought about that, but I guess it is.

Automatic chargers are designed to keep batteries fully charged.

I have left my TT plugged in for the past 3 years at home (except when travelling/camping of course), you have to keep up with the water in the battery.