How to attach short bottom plate for door frame to concrete floor?
You could run your PT plates across the doorway to begin with, then cut them out after the fact. (Predrill the plate and use tapcon style screws and you won't split even a short bit of plate.) The upside of this is that you'll have a straight line across the doorway. Build a wall on this with another bottom plate/studs/top plate and it's easy to plumb it.

Alternately, you could skip the plate and attach the framing to the wall, but that could have issues with the lumber running nice and straight in line with your opening. Best practices would have you use pressure treated lumber in contact with concrete. (It's a bit of a waste of lumber to have 4 or 5-2x4s stacked up where you don't need them. Plus you'll have a hard time putting a light switch there.)