The 1950s called and wants it's Catholic mother's back!
I know a lot of Catholic mothers and they are nothing like this!
I think Your MIL wants to get your DD into a Catholic Primary and will need a Roman Catholic Baptism Certificate to enrol.
"I don’t think MIL sees this as a full baptism — more like a blessing or dedication. But she is calling it a “proper” one and acting like ours doesn’t count ... at DD’s christening and [she] smiled her way through it. If she thought it was all for show, why bother?"
Your MIL is being extremely disengenious here. She's attended a full Church of England Christening, complete with Godparents of your choice, which makes your DD a member of the Cof E church, able to apply to a CoE school.
I think MIL is calling her RC ceremony a "proper" one - because that's what she's intending, a proper RC Baptism. But she's describing it to you as "more like a blessing" because she's hoping you won't catch on until the ceremony is underway. She smiled her way through your Cof E Christening because she had already plans to do this. You might have even found some Catholic Godparents primed and ready to stand up.
Speak to the Vicar who christened your DD before you speak to Fr Liam so you can get the actual facts on what each of these ceremonies mean, as there's lots of disinformation on this thread including confusing RC Baptism and Cof E Christening
I really doubt that Father Liam has any idea that your DD has already been Christened a member of the CoE church, as MIL's probably described the original ceremony as "more like a blessing" to him too. So it's probably not his fault and is not something he'd want to be in the middle of.
The clerical community is not that big and have lots of ecumenical ties, and they may well know of each other. MIL might have thought to persuade you into having the RC baptism, because the family (MIL) expects it.. but having spoken to both your Vicar and Fr Liam you will be well briefed to say no to this.
Father Liam will be able to confirm if MIL did in fact book an RC Baptism or just a blessing. I think its really important to establish that before having any kind of disagreement over it.
If it really is just a blessing, then you can relax a bit and just insist that you are not changing the child's name. There won't be any need to fall out over it, it will just be a rather nice blessing of your child, and you will have been well warned in advance to stick to your boundaries with MIL. But make sure you know the facts before you lay down the law with MIL and all his family or you could be starting something, which if you were more tactical, could just be quietly managed instead.
BTW... as your DH comes from a very religious family, Did you marry in Church?... CofE or R Catholic? Its surprising that MIL didn't say anything when you first mentioned a Christening.
Your DH was brought up by a staunch Catholic mother, so he must have some knowledge about these things drummed into him.