No, because with a leaky membrane, the proton gradient cannot be maintained
The cell membrane is composed of a protein lipid bilayer. The detergent is very efficient to clean out portions of lipid from the protein part, thus disrupting the cell membrane. In presence of low concentration of detergent, the small portions of lipid are taken out, leaving the cell membrane structure intact, but with smaller pores which can lead to leakage of several smaller substances from inside.
So these leaky membranes don't allow the gradient of proton to build up. Thus ATP cannot be produced. So the cell cannot respire with detergent present.