Whilst I agree with pps that the loss of her sister may be part of why your daughter won't let you babysit, there are a few other things in your post which stood out to me.
My daughter, for her own reasons, decided to turn parenting into a degree course, she was in competition to prove she is better than I was
This makes me think that you and your daughter have different approaches to parenting. Have you openly disagreed or disapproved of her choices? My mom did quite a lot, saying things like 'I never did that and you turned out fine', 'these new guidelines will change again and I'll be right then' and she really disagrees with my approach to discipline (she smacked, I don't. And I won't let her smack DD either). She perceived me doing things differently as 'competing' with her. She told me once 'I would have been a better parent if I'd have had google as well'. She couldn't see that I was making the choices I think are best for DD, based on the information I have, but saw it as me trying to do things differently to her to 'out do' her. You seem to have the same attitude towards your daughter. Because of all of this I won't let her babysit DD, I feel that as soon as my back is turned she will take the chance to do things her way to prove she's right.
Four times over Christmas I tried to arrange to take him to Circus, to see Santa to go on the Santa Express
This may also be part of the problem. She doesn't feel comfortable leaving him with you, she's told you that. And you keep pushing for it. Stop asking her. Arrange activities you can do with him and her, stop focussing on getting time alone with him. Ask to spend time with him and your DD, let her see that you respect her choices as his parent and she's more likely to come round. I'd be more comfortable with my MiL having DD alone as she doesn't pester me for it, whereas my parents do and that makes me wonder if they will just use that chance to undermine the parenting choices I've made (no smacking for example).
One grandchild of nearly two. He should be the light of my life. I've waited for him.
Throughout my pregnancy my mother was very fixated on 'her grandchild' and everything she was going to do with her once she was born. But she seemed to be completely ignoring my role as the mother, other than me being the person carrying her grandchild. Like I was an incubator. She was the only person who didn't even ask how I was once I'd had DD, which really hurt. I felt like I had done my job of giving her a grandchild and that was all she cared about. It's really ruined our relationship. You also say I don't know where my little pal has gone we were once so close This is what makes me think it's partly about her relationship with you, as your relationship with her has obviously changed. Maybe try inviting her to do things without her son and try working on your relationship with her first. I understand you want to see your grandchild, but she's your daughter, your damaged relationship with her should be your biggest concern, but your OP is all about your grandson, maybe she feels that. Anyway, having a stronger relationship with her will give you more time with your grandson.
I could be very wrong but I just get the impression in your post that this is about more than the loss of her sister, and more about a change in yours and your daughters relationship, I think you need to focus on improving that first and foremost.