The two following rulings for Lurrus of the Dream-Den apply:
You must follow the normal timing permissions and restrictions of the spell you cast from your graveyard.
Once you begin to cast the spell, losing control of Lurrus won’t affect the spell. You can finish casting it as normal.
By the second quoted rule, I see if I don't respond to Lurrus being cast and entering, and my opponent chooses to cast a permanent from their graveyard, even if I then kill Lurrus, the chosen permanent will still be cast.
So my question is if my opponent casts Lurrus and it enters the battlefield, can I immediately kill it before my opponent gets priority to cast a permanent from their graveyard.