This is a simple and ultralight (90 gram) harness system for securing various loads to the handlebars while also protecting them from friction damage by straps.
Industrial sticky back Loop Velcro is attached to the bar at four appropriate locations. Two straps of about one metre each in length, with four matching Hook Velcro segments are wrapped around each side of the loop, with a full wrap at each location. The full wrap makes for a very secure mount and also moves the load forward by roughly the diameter of the handlebar at the stem side of the loop. A rectangular piece of heavy fabric, in this example scrap packraft flooring, is bar-tacked to the webbing. The pictures below show a system built with 19 mm , 3/4 inch, webbing. Total weight ~90 grams.
Materials: 2 metres of 19mm strap from MEC <C$3, 2 pairs of Nexus Air Lock buckles from MEC C$3, 2 D-Rings to be sewn to the loose end of each strap C$1.50 (optional), Velcro Industrial Strips C$15 (Enough to do a few bikes and solve other gear mounting problems.)
Sea to Summit, Seal Line, Outdoor Research, and others all have various weights and sizes of bags that may also meet your needs. The MEC “Nano” series of dry bags are not recommended as the materials are too light for sustained use.