Yikes! I'd come down on her like a sack of bricks, underlying issues/SN or not. There's no excuse to act with that much disrespect towards you.
And... SN kids aren't blithering idiots, they may struggle in certain areas like impulse control or organization, but it doesn't cause them to disrespect you or deliberately misbehave like she is. You may have to adjust your parenting a bit to help set her up for success though with a stricter routine, stricter boundaries, and clear consequences.
Do you have clear expectations of behavior and consequences or do you just make them up on the spot when you lose your temper? Do you and DP shout at each other or back at her (and reinforce the notion that's how you behave when upset)?
I'm not sure cancelling a birthday party is a good ultimatum. If she is struggling to control herself, a punishment that far out is going to be lost on her. Also, celebrating your birth shouldn't be dependent on behavior, otherwise it gives the undertone of "I only feel blessed to have you in my life so long as you conform to my rules". You don't want to teach her that love comes with stipulations, that's not going to give her a healthy outlook on life.
If yelling is the problem, disengage every time she yells. Tell her calmly "We do not yell in this house, we may continue this discussion when you have had a moment to breath", then walk away. Continue to remind her every time she shouts. Don't make it about having the last word or having her submit to your rules. Your biggest goal is repairing your relationship, and as the child she will need your leadership to do it. Then when she's calm, tell her that her behavior is not acceptable in this house and what steps she can take to stop it and what consequences she thinks will be good to help her stop. You might be surprised. Sometimes kids have creative and interesting solutions to their problems and honestly, they have harsher consequences than I'd have in mind when I ask. Then hold her to that.
Make sure your DP is on the same page, because if you both have different expectations she's going to play one against the other.