N.R.
I would not let her first time being away from you start at the Gym. No matter what the staff says, kids are not their first priority.
Find a family friend whom she knows, and will go to. It is ok to take just one child to the gym with you. Also, remember to wash your son's hands after leaving the gym daycare. I have 5 kids, 2 of which area not yet in school, and mine would come home with so many sicknesses, that I found an alternative place to leave them when I needed the gym.
good luck and remember it isn't wrong to want to be with mommy. You only have 18 years of that anyhow..