*I'd put the ball in the other mum's camp. I'd say something like,
"So, it's DD's birthday party next week, and DD wanted to invite all the girls, but now she's a bit anxious about inviting your DD because she's worried that her toys are going to get stolen again.
I really don't like the idea of leaving one child out, so I wondered if you had any thoughts about how we could make sure this doesn't happen again."
If she can't come up with a solution that satisfies you, DD doesn't get an invite.*
I agree with this. If S’s mum couldn’t think of any ideas I’d probably suggest she comes along to keep an eye on S.
Alternatively, I’d be having a serious word with S at the start of the party myself about how taking other people’s things is not ok, you know she did it before, you’re giving her a chance to show she won’t do it again.
It doesn’t sound like S is horrible or badly behaved in any other ways so I would hope a bit of a stern word and knowing another parent is aware of the previous stealing would be enough to nip it in the bud.
I wouldn’t leave S off the invite list, nor would I be encouraging my own daughter to turn against S and decide she is awful (as you have fairly clearly been doing OP).
Totally agree the girls “sorting it out between themselves” is unacceptable.