I think that consistent, upbeat honesty is almost always the best policy. You don't want to do anything to shake your daughter's faith in you or in consistency.
Something that worked well with my daughter when she was going through a clingy stage was to hold her, introduce her to her new caretaker, strike up a friendly, low-impact conversation to show I trusted them, sit with my daughter for a few minutes while she got involved in a toy, and then tell her calmly but positively "Bye, I'll come back soon."
Since your daughter doesn't yet understand "come back soon," you might do well to prep her for a week or so by practicing ever-increasing periods of you "leaving." Start with peek-a-boo games at home: hide behind a book or blanket and tell her you'll come back soon. Then allow longer and longer times before you pop up again. Extend that to leaving the room for longer and longer periods. And do this in different rooms in your house, or in your yard, or other people's homes, so she learns this is true everywhere.
She will almost certainly become blase, or even bored with the game, and get involved in something else. But she'll learn the phrase and get a sense of what it means. And she'll develop faith in the fact that you will, indeed, come back soon. When you take her to the gym, you can start by repeating the game that has by now become familiar. Plan to give it at least 15 minutes before you head out for your workout. You'll probably notice that she relaxes into her new environment fairly quickly. If her first few visits are too stressful for you both, make your first few workouts very short, and try again in a day or two.
But even if she does cry, remember that this is more the norm than the exception. I like the term "Be as inevitable as the tides." Bazillions of anxious mommies have to leave bazillions of anxious babies crying in the arms of a bazillion caretakers when they go to work. In almost every case, the trauma of separation is temporary – I used to babysit a neighbor boy who would scream as though tortured when his parents left. Two minutes later, he'd be happily playing. His mom always had to sneak back and look in a window to assure herself that her son was okay.