I think you should bear up and let Grandma take her for an hour or so. You may not be ready, but your baby is probably more ready now than she will be in a couple of months when separation anxiety kicks in. As others have said, it will be much better for both of you if there is someone she will go to happily besides you and your DP.
I am concerned, though, about you not being comfortable leaving her for more than 15 minutes, even with her father. Unless you have reason to believe he is an unsafe parent this seems an unusual degree of anxiety, especially as you describe him as involved and keen. I think it's worth a discussion with your health visitor about how you are feeling and why your invisible umbilical cord (we all have one - mine is stretched to its limit just now with my oldest at uni!) is so short.
Unless you don't think your MIL is going to be a loving and safe caregiver, I think you should be brave and let her take the baby out for an hour or so between feeds, or leave them both at yours while you go out for coffee with a friend or get your hair cut or something. A warm and close relationship with her grandparent is a wonderful gift to give your child and I would invest in ensuring that their relationship starts at the right time for both of them, which is probably now for your MIL and for your baby as well.
GPs and dads often don't feel they can bond with the baby when mum is there - they need to feel like they are the one giving care and are not being watched and judged on their baby skills. You have to let dad put the nappy on backwards and let Granny sing Cliff Richard to her. Your baby will be even happier with more love and care from more people. And you may find that you get used to these sessions and start to enjoy an hour or two alone, or with a friend, once or twice a week.