Jewish dilemma

As we know, mushrooms in the Lactarius family “give milk” when their gills are cut.

Does the milk-like latex mean that the lactarii must therefore be “dairy-like” and, in accord with kosher dietary laws, should not be eaten with meat?

Or, should we reason that since mushrooms belong to the fifth kingdom and are neither dairy nor meat, they are “parveh” and can be eaten with meat?

What do you think?