The short answer is that you cannot get bare-earth data straight out of MME processing. Anything sitting on the ground, including vegetation, is going to be included in the "surface".
You can take the products produced by MME and then edit the point cloud in some other software to remove any data you do not want included when you generate contours.
For capturing data in thick trees or generally homogeneous surface, you need really good overlap at sufficient altitude above the tallest objects. Lawn-mower pattern is fine.