Hello Nick.
I just read your message this morning. I would be hesitant to try to
transplant the tree. Keep in mind that the tap roots of trees often
extend as far down into the soil as the shoots extend above. Also,
if the tree has been infected with Cryphonectria parasitica,
("chestnut blight"), you may unintentionally help spread the blight
fungus. With that in mind, what, exactly, do you mean when you say
that the tree is "sickly looking"? Look along the trunk for orange
spore producing structures and/or cankers, especially in any cracks
in the bark. Most American chestnut trees of the size you are
describing have not yet begun to fissure. It (the bark) is generally
gray and smooth. Cracks may be initiated by insect/woodpecker
activity, damage by contact with adjacent flora, or damage by fallen
limbs. These cracks "open the door" for C. parasitica to infect the
young tree.
Also keep in mind, should you decide to transplant the tree, the area
in which you would move the tree to. In our research along the
Highland Rim of Kentucky and Tennessee, we are finding most specimens
on dry, well-drained, and relatively acidic soils. Although we have
found a few specimens in low lying, moist soils, they do not seem to
thrive. By this I mean that we do not find clusters or "pockets" of
trees. Those we do find seem less able to fight the blight fungus,
perhaps due to the increased moisture. As well, we have found
no "large" chestnut trees in these areas. In short, if you do
transplant the tree, it might be wise to place it in a relatively
dry/well-drained area and follow that by utilizing a relatively
acidic fertilizer. Also, if you do transplant the tree, wait until
the leaves have fallen and the tree is dormant. This will reduce the
initial environmental stress upon the tree.
-THRASHMIKKI