I can't say that it is the 1918 prototype or the 1919. Things fail to match both ways. The heat shield cutouts style was used on both, apparently, but is rare. That the heat shield appears to taper into the receiver housing isn't something I can find on any other images. If the gun is a production Browning, then finding what is apparently a rare variant is not likely to be as simple as going down to the local machinegun rental place, I am afraid.
"If you look through your scope and see your shoe, aim higher."
-- said to me by my 11 year old daughter before going out for hogs 8/13/2011