Hello Bcrdnk, I had a 135 that would stop like you describe. The trouble was the fuel cock at the tank had a tiny fine filter on it( can't be seen by looking through the cap) that gets blocked by crap in the botom of the fuel tank sloshing around, causing an intermittant fuel starvation.
Due to lips on the fuel filler and fuel cock holes I was not able to clean the tank fully.
My solution was to remove the tiny fine filter on the fuel cock, and when I heard the beginnings of fuel starvation, stop and blow (disconnect fuel cock line at filter end) back (remove fuel cap first) through the fuel line to clear the blockage and carry on. Also I avoided going below a 1/4 full, this seemed to help.
Good luck, and let us know what the fix was thanks.