What amenities are needed? In my opinion, Brantley Lake SP is the best of the 'developed' campgrounds around Carlsbad but, if you don't mind dry camping, Dog Canyon in Guadalupe Mountain National Park is a great alternative. It has water and flush johns but not much else other than location and the fact that you'll likely have the place to yourself. Security isn't an issue as the CG is in a box canyon and the road in runs right by the resident ranger's house.
When visiting the Caverns, get reservations for the Lower Cave tour. Its an extra cost but a great way to see parts of the cave most folks miss.
Lincoln National Forest is adjacent and makes for a beautiful drive from Dog Canyon to Carlsbad. Sitting Bull Falls is in the NF and well worth a visit.
The Living Desert Zoo and State Park is located in Carlsbad and a great place to spend at least a half day.
The Pratt Cabin in McKittrick Canyon is a nice day hike but be sure to take plenty of water.
Its a beautiful area - have fun.