I'm 50/50 on this one, I agree with what other PPs have said regarding your MIL, but I also agree with torain and jings on this.
I agree with the PPs saying that if you're not comfortable leaving your DD alone with your MIL right now, you absolutely have the right and it's not unreasonable either to tell her that you just aren't ready. You mentioned she's been manipulative in the past, try and frame what you tell her in a way that she can't twist it into anything but what it is. Like you're really happy knowing that your daughter has a wonderful grandmother that loves her and that you sincerely appreciate her wanting to bond with her, but you're quite attached and are having a hard time being away from her so you're not ready yet and you'd be worried that it would hurt her more if you were to drop her off only to end up picking her up 15 minutes later due to your anxiety about not being ready. Basically "I'm doing this for you" in a way - it tends to work with overly dramatic and/or manipulative people.
As for what torain and jings said however - you do sound like you have seperation anxiety. When I read that you won't even leave your daughter with your husband made me really sad for all three of you. Your daughter wasn't conceived alone, she has two parents and is very fortunate to have both of her parents not just in the picture but living together. Her father deserves time to bond with his daughter just as much as you do, it's not unreasonable to not allow your MIL alone time with DD, but it IS unreasonable not to allow her father alone time with his daughter. Try and empathize/put yourself in his shoes, imagine the situation reversed and think how heartbroken you'd feel if your husband wouldn't let you spend more than 15 minutes at a time alone with DD. I think it would help your anxiety, make your husband happy and help set your DD up for success when it comes time where you HAVE to leave her in someone else's care if you were to talk to your husband about how anxious you are about leaving her and that you want to work on it together by having him watch his daughter for more than 15 minutes at a time. I suffer with generalized anxiety disorder / panic disorder and when I feel that extreme anxiety I have learned that I have to sit with it and then let it pass over me. The first time you leave her for longer than 15 minutes, it may only be for 30 minutes, but the point is to keep gradually increasing the time you spend away until you're able to go out for an hour or two, spending some nice time doing whatever you enjoy without feeling the need to run home because you'll know by then that your DD has two amazing, loving parents that will always take great care of her.
For your own mental health as well as the benefits to your husband and daughter I sincerely hope you consider trying to increase the amount of time you can spend away. Baby steps are fine, increasing your time apart by increments of 5 minutes once a week is a workable goal, I think you're more than capable of doing it.