Yes, you can leave your RV for the day. We left our RV to visit White Sands National Park.

You are allowed to per BLM rules (links below). I saw rigs left for the day when I visited in September. I was not here during peak season, but everyone was very respectful of each other. The general rules for BLM NM says don't leave property unattended for more than 10 days, but I personally wouldn't leave anything for more than a day or so. https://www.blm.gov/programs/recreation/recreation-activities/new-mexico https://www.blm.gov/programs/recreation/recreation-activities/new-mexico

We did. No problems.