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Yeah, that's probably a better approach than using the energy of the orbits.
Okay, so if we set the weight to 1kg, force is rR~e~^2^ - GM~e~/r^2^, where M~e~ is the mass of the Earth and R~e~ is it's rate of rotation, which is a low number in radians per second. The antiderivative along r is then -1/2r^2^R~e~^2^ - GM~e~/r, but you actually don't need that because you just take the derivative again to find extreme points. rR~e~^2^ - GM~e~/r^2^ can be restated as (r^3^R~e~^2^ - GM~e~)/r^2^, and that numerator is a diverging, increasing function as you move away from 0, which means yes, the energy is unlimited.
Welp, I was wrong. I think the trick here must be that the rotation of the Earth didn't actually come from the gravitational collapse itself, but from the pre-existing inhomogeneities of velocity distributions in the early solar system. Even if you could slingshot the mass around the sun, back into the Earth, and collect it again, you would somehow transfer enough of Earth's angular momentum back to the sun to offset the energy gained.