Right, English lefty liberal practising RC Catholic here, who would never do what your MIL has done, but I did baptise my own children and just signed one up for First Holy Communion lessons.
I think the issue is you are all speaking very different theological languages and actually your CofE vicar and your husband are the most at fault here. (It is possible Fr Liam also is, but at this point in the game we don't really know what he said, what he is offering and in response to what information he has been fed by your MIL). Your MIL is being a dick too, and it is inexcusable and she needs to be told no, but it is unclear where the tangle of theology here came from...
I'm actually going to suggest that your CofE vicar started the confusion by suggesting you could have a 'welcoming ceremony' that wasn't very religious, and then in fact did a full-blown baptism (which is the same thing as a Christening theologically, they aren't different ceremonies in either the Anglican or RC Catholic church). I appreciate Anglicans can be a bit fluffier, but this borders on dishonesty by the vicar in my book. There are quasi-legal ramifications to a baptism certificate regarding church/national records and school admissions. It is significant theologically, legally and politically, no matter how you try to dress it up, or even if that wasn't what you actually wanted.
If your RC MIL is half the Catholic you say she is, she would have seen right through this. Was she actually at the ceremony? I think there is a risk that your description of what was actually a full blown baptism/religious ceremony as a 'welcoming thing' may have convinced her it wasn't actually a baptism/christening of any religious significance, because that is how you described it, particuarly if she wasn't there. Her wanting to baptise your children without your say so is obviously not ok, but its not the same of a redo, as quite frankly a Catholic, including myself, wouldn't see what you have described as a baptism (even though, it very much was... see my comments about your vicar above).
I get your DH isn't that fussed about the religion he was bought up in, but I am shocked as somebody raised by this woman doesn't seem to understand anything about religious sacraments and what a baptism actually is. He really needed to be translating between his MIL and you a bit more than he has bothered to, and quite frankly, its bloody lazy of him, regardless of his attitude towards religion. Where is FIL in all this? Did DH get confirmed? Did he attend a Catholic school at secondary level? This would have been very basic stuff.
Then... onto Fr Liam. We don't really know what has happened there but FWIW I and my non-Catholic husband had to attend a month of baptism classes, where we actually had some of the aforementioned theology explained to us (for the nth time for the RC Catholics in the room) and it was both implicit and explicit the expectation that we were a Catholic family committed to bringing up our child in the Catholic faith including commitment to further sacraments, which brings with it Mass attendance. The baptism ceremony is heavily focused on the parents and godparents, it is very odd that you would be excluded from this, as you are the ones raising the child. Similarly, priests are indeed very careful about treading on another priest's toes.
Then onto the concept of a 'blessing'. This isn't quite a neutral as it sounds, though sneaky priests (a bit like your CofE vicar) might try and make it sound benign. It is not. If you want a Catholic marriage blessing during or after an Anglican marriage ceremony for example, you still have to promise to bring the children up Catholic etc, in exactly the same way you would if you had a Catholic wedding ceremony. I would guess that a Catholic baptism blessing might involve the exact same promises, meaning, it isn't really a benign 'blessing' at all. However, I have never heard of such a thing for a child before, as adults can convert, but not children.