Following on from my previous reply, I'd suggest the following.
Tell your mum you're not interested in having any conversations with her about weight, and that you won't even be telling her your baby's weight when she is born because you're finding her repeated comments about your friend's baby's weight so fucking annoying you don't want to be subjected to the same thing times ten about your own baby.
Every time she mentions anybody's weight, in any context, say, "Mum, I told you, we're not talking about people's weight. I think you have a lot of emotional issues around weight, I had an eating disorder myself as you know, and I am determined to break this toxic cycle and not pass on any hang ups about weight to my daughter. So we're not talking about weight."
If she continues, you end the conversation. Try changing the subject first, if she insists, you hang up the phone, stop responding to texts, or physically walk away from her.
With regards to other people, make your baby announcement without any reference to weight. If anyone other than your mum asks how much your baby weighs, tell them and don't mention your mum.
If your mum then says, "Sandra mentioned that Isla weighed 8lbs when she was born" (which I think is unlikely because Sandra will think she already knows her granddaughter's birth weight and she's unlikely to want Sandra to know that she isn't privy to this information, just repeat, "Mum, I told you, we're not talking about people's weights."