Air cooled guns had the advantage that the cutouts in WW1 tanks for these guns were much smaller than those for water-cooled guns. Another thing is that the barrels of the water-cooled guns were exposed to enemy fire.
As far as I know the Hotchkiss is heavier than the Lewis and so was chosen for infantry and tank use while the Lewis was preferred for aircraft use. In my opinion it would have been a problem to handle a Lewis guns (including the big ammunition drum) inside a WW1 tank. Perhaps there were only production problems to chose the Hotchkiss instead of the Lewis.
Perhaps someone else knows more.