Do you have newspaper that you can shred and "stir" into the heap? I've used shredded newspaper as a brown many times, due to the irregular availability of leaves, etc. And I do mean shredded--through an office paper shredder!
Placing a cover over the heap needn't necessarily mean depriving the heap of air: A painter's tarp (canvas) as opposed to an outdoor tarp (plastic/oilcloth) will allow air to pass through, while preventing most of the rain from penetrating the heap.
It's a little late in the year, but wood ash from fireplaces/wood-burning stoves will also help restore balance to a compost pile. My girlfriend used to use her fireplace for house heat but now has a pellet stove, and I've lost the wood ash I used to get from her.
El Cerrito, CA
USDA Zone 9, Sunset Zone 17