Can I use Caldari LP to buy a PNI from State Protectorate Corporation (which I am guessing is Caldari too)?

Just curious whether I can earn 1,000,000 Caldari LP from all the different Caldari faction corps and spend it at the State Protectorate Corporation which I think is also Caldari. I haven’t done LP in a while.
I desire the Phoenix Navy Issue BPC.

No, you need to earn 1M LP from the State protectorate (or through concord)