I can't do that, especially not in the house.
I can't do that, especially in the house.
Are both of these sentences grammatical/acceptable? What is the difference in meaning between them?
They actually have the same meaning. In your first sentence
I can't do that, especially not in the house.
Especially is used to emphasize the prohibition.
I can't do that, especially not in the house.
means when not in the house, it's even harder for me to do that.
I can't do that, especially in the house.
means when in the house, it's even harder for me to do that.