A larger diameter bullet may or may not carry more energy - it depends on mass not diameter. the formula for kinetic energy is half mass times velocity squared. Velocity has an exponentially larger contribution to energy than mass. This isn't reflected in an exponentially different wound.
Your statement of "all else equal aside from energy" implies same mass, diameter, construction, spin, etc - just different velocity.
In a scenario with 2 identically constructed bullets at the same velocity, one with greater mass and greater diameter - the larger bullet will create a larger wound channel but the evidence suggests that this isn't as pronounced as you might expect. The additional diameter does create a slightly larger wound channel and more mass = more fragments. Energy has little to do with it.
The best evidence we have to test this is relative gel tests. They indicate relatively small differences in max permanent cavity diameter.
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