Nevermind how I intend to clip through the surface. You could say I intended to drill a tunnel kind of like the drop through the earth problem.
I know how to do this calculation assuming all of earth's mass is at its center, and this yields the ridiculously low delta v requirement of 460m/s (velocity of earth's rotation) to drop PE to zero relative to the core, an infinitesimal amount to raise AP to 1.5 million km, and 515.7m/s (Vorbit = sqrt(G * M / R)) to establish orbit. If I ignore Oberth and other stupid tricks altogether I get the simple answer of 7207 m/s (velocity required for LEO - velocity of earth's rotation) to increase my eastward velocity adequately.
But the earth isn't a point mass so that calculation is stupid wrong. I can't get infinite Oberth from being at the center; however the effective Oberth value is still very large, but I can't figure out how to estimate what it is.
[Deliberately ignoring moon assist and other such maneuvers; that's not the point of this question.]
So I've been working on this and realized I made a mistake; I accidentally assumed circular orbits where I had intended only orbits completely clear of the ground. Thanks to a tip from Cosmas Zachos, I now think I asked a question I am unlikely to recall enough calculus to understand the answer to.