If you were just wanting to slightly darken it, I would think it wouldn't take much tea water to do that.
If you have access to a clothes line, spray it while it is hanging. A gallon of water and 3 tea bags would go a long way if you sprayed it on. The tea shouldn't hurt the grass any either. Using the same water throughout shouldn't make it splotchy, as it is going to run off, and not pool if it is hanging up.
Soaking it, it is going to try and float, the folds and creases could get less/more dye water depending on how it is laying in the water.