No, sorry, I think you are being perfectly reasonable! Agree with Grovee and Cupcakes aka BupperFlaps 
Your mum appears to be trying to show you very clearly that she knows what's best for you and your child, thankyou very much. That you may be acting like a silly little madam but that she will override this and make sure all's ok in the family...GRRR!
'my mother doesn't see anything wrong with the cruel way that my sister has been treating me and isn't prepared to take her to task for it.'
- She doesn't have to take her to task for it, it's not her argument - but she sure as hell has no right to dismiss it as unimportant.
'My sister has had several opportunities to see DS with me present - I have invited her to spend time with me and him so we can work out our differences but she hasn't bothered. Instead she is going behind my back to see DS.'
That settles it for me. Doesn't sound as if you've been unreasonable wrt your son's relationships here (though to be honest if you were it would still be your decision - he is YOUR son, YOU are responsible for him ) - it's more that your mother wants to be the one putting you in your place, so to speak. I don't like your mum's role here. Is she usually a bit of a controller and a stirrer?
Screaming abuse at you in the street?
I think your first post sums it up just fine:
'she can't take take him over so my sister can have a lovely time with him yet still gets to carry on treating me like shit'
If she's treating his mother like shit, then it's best she doesn't see him until she can accept your offer to come over and settle your differences. Yes, that's right, YOUR offer. That YOU are in control of. Nothing to do with your mum.
Sounds like this offer of babysitting has more strings attached to it than is helpful in this situation. Tell your mum that if she wants to help, rather than flexing her muscles, she'll urge your sister to come and see you. Until then, tell her to BUTT OUT of your adult relationships. And tell her that you think it's best she doesn't babysit if she can't respect the child's mum's decisions.