Sprout, if you haven't read Dr Gordon Neufeld's book Hold On to Your Kids, I highly recommend it. You sound like you're doing all you can. She is clearly distressed and time with you guys is likely going to be the cure. At the moment, she is crying because she is upset and needs comfort, so go for it. You are a savvy enough parent to know when/if that need for comfort changes into a manipulation strategy.
Some of our kids these days are just really sensitive little beasties and need lots and lots of connection and face time. When you HAVE to leave her, make sure you bridge the gap by telling her when you will see her next and that you'll be thinking of her etc. All the stuff I'm sure you're already doing. You might want to give her some little talisman of yours to hold on to and keep safe til she sees you again. It helps to maintain that connection to you and will help to soothe her a bit if that's one of her issues.
She's only 3. Cuddle her, read to her, play with her and help her over the bump. She's not manipulating the situation at the moment. You'll catch that if she crosses that hurdle. Its absolutely OK, and even encouraged to comfort and snuggle the stuffing out of her right now.
Good luck Sprout. I completely understand your struggle here. Parenting is a delicate balancing act and you have to do it without a net!