One way to look at this is that your MIL has been brought into your life so you can learn how to become politely assertive, which is a great quality that you can pass on to your children. Kids usually learn this skill from their parents, they don't automatically know how to do it!
Your MIL uses passive-aggression and manipulation to try to get what she wants (ie she's covertly controlling), which is a very unpleasant behaviour to deal with. What she's doing is a form of bullying, and as with all bullies, the only way to stop it is to make it clear that the bullying behaviour won't result in her getting what she wants – but good behaviour will. (Up to a point you're comfortable with, of course.)
First thing is to get your husband firmly onside, because as you've seen, if she can't get her way with you, she'll try to manipulate him. So you need to have a serious talk with him that her insults are hurtful to you and making it hard for you to have a genuine relationship with her. There is literally no good reason in the world for someone to make insulting comments implying that you're antisocial and unattractive. So that disrespectful shit needs to stop.
If your husband is willing to dismiss it with 'she's just joking' then your problem is bigger than your MIL. He needs to understand that insulting someone isn't a joke, it's a method of establishing power, and it's not okay. If he grew up with this kind of thing, he might take a while to get his head around it. Once he's on board and recognises that it's not all right for his mother to insult you, you need to take a united front on a boundary around this behaviour.
The next time she says something like her previous comments, take a deep breath and say, "Kathy, I feel hurt and insulted when you make negative remarks about me but directed to the baby. I would like you to stop doing that."
If she says "Oh, I'm just joking, don't be so sensitive", say in a calm tone with a pleasant expression on your face (you may need to practice this), "You might find it funny, but I don't, so I'm asking you to stop. Having a good, close relationship with you is important to me, and I don't enjoy spending time with you when I'm being insulted, even as a joke." Here, you're offering her a positive outcome of playing nice and also showing that there's going to be a consequence if she doesn't play nice. She's a master of passive-aggression, she'll get the point.
(In the event that she suddenly becomes capable of addressing issues directly, and says, "What are you saying, that you'll keep the baby from me if I don't pander to your sensitivity?" you can reply, "Yes, I'm saying that I feel reluctant to spend time with you when it involves being insulted.")
At this stage she might grumble a bit or say something passive-aggressive like "We hardly get to spend any time with you anyway" or some other kind of bait; literally just continue to keep a pleasant expression on your face and don't say anything in response. Calm silence is a powerful tool of assertion. Your goal is to communicate your message calmly, clearly and politely without giving an inch or allowing yourself to be caught up in a drama.
The behaviour might well now stop but if it occurs again, you will have to make the boundary crystal clear. Again with the pleasant face and calm tone, "Kathy, I've told you before that I don't enjoy spending time with you when you insult me through the baby. If you continue to disrespect me in this way, I'm afraid I will have to end our visit and take a break from seeing you."
This may spark a lot of protest and argument. "How can you keep me away from my grand-daughter! I can't believe you're being so ridiculous! What nonsense! What gives you the right? What's your problem?" Etc etc.
Again, you need to stay calm. She's freaking out because she can see she's in danger of losing something she values and she's resorting to her standard tool of control, which is insults. But this is just noise and attempts to get her way through escalating drama. You need to rise above, ignore her bait (ie don't respond to the specific things she's saying), but instead stay on message, repeatedly saying the one thing that reminds her of the positive outcome of behaving well and asks for an agreement on better behaviour from her. "Like I said, Kathy, having a good relationship with you is important to me but that's just not possible when you insult me through the baby. Can you please stop doing that?"
When she sees you're determined to hold the line, it's likely she will back down (because you are the one with the real power here), although she may well serve the backdown with a side of insults "Well, I think it's ridiculous, but fine, if you're going to be that way." An agreement is all you're after, so again, ignore the bait and say, "Great, I'm glad you understand."
Now you have a line in the sand. Bullies are usually quite weak inside, despite their external bluster, so it's quite possible your problem will be solved. However, if she keeps doing it, the very next time it happens you have to deploy the consequence immediately. She insults you, you pick up the baby and say, "I've been very clear with you that I won't tolerate insults if you want to spend time with me and baby. It's time for you to leave." Then open the door and stand by it waiting for her to leave. (Or if you're at her house or somewhere else, "It's time for me to go." Then do it.)
Your husband has to be 100 percent behind you on this. If she goes to him to complain, he has to say, "Mum, the way you've been speaking to Naggy is really not okay. She doesn't like it, and I don't like it. I know she wants a good relationship with you, so if you can speak to her more respectfully, everyone will get along."
Sure, you could try to pass this to your husband to deal with in the first place, but you won't gain her respect that way. Most bullies actually have a grudging respect for people who are strong enough to refuse to be bullied AND can do it in a civil way. Game recognise game, you know? Of course, a small number of bullies don't, and will continue to escalate, but that's when you know you're dealing with someone truly toxic, and have a genuine problem on your hands. I'm sure you'll make another thread if that happens.