Your conclusion is right but your argument appears to be lacking.
> You integrate this over the path that your center of mass takes, which is going to change linearly with your scale k (d).
But if that's how you define d, then it is the height h! Here you are assuming that d scales linearly with k. Later you are saying that h is invariant with respect to k. Which is it?
Jumping is impulsive, not sustained, so your force times distance formulation doesn't seem appropriate.
Thompson has a nice analysis in his classic treatise On Growth and Form, which is all about dimensional analysis applied to biology. Here is the relevant excerpt: