I am going to go back a step and say I don't agree with this. It's not that deaf babies' brains don't work the same way, it's that their ears don't work the same way. In the cognitive sense, bilingualism is the same for a hearing child or a deaf child. Some people grow up bilingual in ASL and LSQ (Quebecois sign language) for instance, and are bilingual in the same way as someone who grows up with English and French.
What's different is the medium through which the input (words, phonemes, grammatical structures) enter the brain. That is why so many people in the ASL community, and in the Linguistics and Cognitive Science community, support ASL as a first language for deaf babies, because it's completely accessible visually. But you already know that.
Anyway, my point is that it's not that deaf children learn language differently, it's that they learn speech differently. I do think that you're confusing them here to some extent. And that's why I think the research about bilingualism for hearing children is useful. I think that the person who told you Miss Kat can't be bilingual hasn't done enough research into bilingualism in general, and may have general misconceptions about it, as many people from essentially monolingual societies do.
On a different note, I'm also a hearing parent, and I will say that I think we are all a little audist, despite our best intentions. Well, maybe there are exceptions, but while I thought I would be one, now that I'm actually facing these issues, I can see I'm really not. We want our children to have as many opportunities as possible in life. I would like to think that it's absolutely fine for my daughter to sign and not speak, but at the end of the day I don't believe everyone in my community (or my family) will learn to sign. I do think this comes from my natural bias as a hearing person. It is hard for me to step out of the hearing way of thinking, and part of that is that I want my daughter to hear as well as possible. If she ends up not being able to hear even with technological assistance then I will rethink that, but for now I am being honest with myself and saying that I would like her to be able to speak well. I also want my hearing son to speak perfect English even though we live in a non-English-speaking country, because I think English is a valuable tool in life. Why is it so valuable? Because of the imperialism of English-speaking countries. So yes, don't know what the word is for that, but let's call it Anglicist. Should we try to fight for society to be more egalitarian? Yes. Should we do it at the expense of our own children's happiness? Probably not. But as for figuring out what will make our children happy...well, that's up to all of us individually, isn't it? And we all have our biases. Mostly we all just do our best and hope we're right.