I assume that the antibody recognizes and attaches to the protein from the outer cell membrane. So, maybe there's no need for permeabilization?
Bonus question: can we run sequential staining for one sample? The first step would be staining the sample for non-intracellular proteins (and without permeabilization), and in the second step, we would run the protocol with permeabilization. Does that work?
To answer the first question, no if the antibody recognizes posterior (outside of the cell) domains of your protein but yes if it recognizes domains found on the interior of the cell membrane.
For the second question: in flow cytometry, you can first stain membrane proteins of interest and then you permeabilize cells and stain the proteins from the inside of the cells. This should work here too.
