Battery Advice, need to buy a new one

My wife and I just bought our first camper a few weeks ago, a 2004 Flagstaff 176 Pop-Up. I'm pretty sure the battery is original, the previous owner always camped where there were hookups and didn't know if it would hold a charge. It wouldn't charge when the camper was plugged into AC power for several days, and it also wouldn't charged when hooked directly to a battery charger, so I'm pretty sure it is bad. I want to replace it but not sure of the best kind to get. I know some can be damaged if discharged past a certain point. Are they all this way? Can some be discharged father? I don't mind spending a little extra if it will have more capacity as I do plan on dry camping.

Also, I might consider getting a second battery, however I don't have room on the front of the camper for another battery, (pretty small camper). Would it be safe to transport a second battery inside the camper? It would be in a battery box and secured, then removed from the inside once we are camping.