OP, whether or not DSD can or cannot use her knife properly, whether or not she is anxious and needing stability or just a brat - your & her dad's approach evidently is not working.
Obviously there are issues with her biological mum you've only touched upon in this thread - some advice, stepmother to stepmother - do not take her head on via the way you handle the child. It only ever ends in tears.
I can appreciate that after 3 days a week of living with someone that doesn't seem to care about her very much that coming into a house where she is expected to behave to polar opposite standards is distressing or confusing for her, can't you?
For any other child in a "normal", stable situation, YWNBU, but I think you're on the borderline here. Either way, surely there is no harm in changing your approach for a month or two and seeing if she starts to feel less stressed.
It sounds obvious but you have to be the peacekeeper in this relationship, her father is undoubtedly not happy about the way her mother treats that and I guarantee you your DSD will understand his opinion all too well. If you back up your partner relentlessly the DSC sees you as the reason for "poisoning" their parent with views so polar to the other - hence the desire to drive you apart and the sheer anger that they won't succeed with that. Try to be neutral, as best you can.
There are presumptions made on every thread, and the irony strikes me that those who call out at usual and hully for projecting stress or upset on to the little girl are also projecting their own notion that she is just a brat and everything's fine onto the situation - because not even OP knows the girl's reasons for throwing a strop and being upset over some chicken.
How likely is it that a 10 year old able to cut up their food would avoid doing so because they could avoid doing so? Especially when you negate icecream rights? Not very, IMO. How likely is it that a 10 year old blowing up over some chicken and expected table manners has other difficulties going on? Quite, probably.
So why not sit down and ask her - gently - "what's the matter? The deal over the chicken, what happened with using your knife?". Let her say her piece then draw a line under it.
(TL;DR version: cut the kid some slack, move on).