I would suggest brush removal/pruning but that seems too labor intensive to really make any difference. I'd say wait till the driest part of the year and accidently knock over a gas can and then accidently drop a lit match..

Seriously though, try planting some shade tolerant forage species in small test plots and see what kind of results you get. I'm loath to advocate you planting non-native species so try to find something that occurs in the area or at least in the region. Exotic species can cause HUGE problems and have rippling effect that can virtually effect any aspect of the ecosystem from plant communities to big game.