Thanks for the quick replies! Attached as requested are 4 more pictures of the trunk detail and canopy. What looks like a small stick across the pool of sap is actually a live root, presumably from the white oak.
The pool of sap smells about the same as fresh cut oak when cut and split for firewood, that slightly "sour" smell that many of you may be familiar with. You can see in the picture that the sap is strong enough that it killed a liriope plant next to it.
I have a modest home orchard, so have a good selection of fungicides and batericides available to me here, including copper based, if any of you conclude one of those might help. Could also, of course, spray with a diluted bleach solution. I'm very familiar with Cedar Apple rust, Brown Fruit rot, and other orchard diseases, but have never seen this "foaming sap" pooling from a tree root base.
In the larger picture, the tree to the left with a dead branch sticking out to the right is a wild cherry, and behind that is a hickory. The original grove consists of several cherry, hickory, white, red, post, blackjack, and Nuttall oaks, and twenty year old redbuds and Washington hawthorns that I planted. Keeping the Chinese privet beaten back from the property border fence is an ongoing battle. Thanks again!