Decide on your crankset first, then select the BB or do both at the same time, a French BB will be of little help, cups and spindle will be wrong as advised earlier Italian is 70mm wide, everybody else is 68.

i have used 3/32" chain on a fixed gear, harder to locate rear cogs but unless you put in big miles, the wear will not be much faster. Long ago the hot Pursuit set up was to run 3/32" chain. I have even used road cranks (Sugino 165mm) with pista chainring bolts, filed off the back side of the crank arm at the spider for clearance, worked like a charm.

Do review the BB drop of the frame to know in advance crank / pedal decisions, most Basso's I have seen are Italian road geometry, too low for me on a fixed bike.