Hi Maggie:
Thanks for your great question!
Planting a new tree under the crown of an existing tree slated for future removal is probably a bad idea as it will be in the "collateral damage zone" of the tree that needs to be removed. Arborists are extremely skilled in their trade, but they can only be expected to do so much.
Additionally, trying to establish a new tree under the crown of an established tree also blocks a lot of sunlight necessary for good growth and vigor of said tree.
Without an on-site visit, it is extremely difficult, if not impossible, to provide you with a good recommendation for minimum planting distance. However, I would strongly recommend that if you do plant a new tree, keep it as far away as possible from the tree slated for removal!
Best of luck!