If you are running 2 ounce copper, I'd take them up to at least 0.2" on both sides for power. Most times it's easier to do this stuff with a copper pour / plane than with a trace. For ground, go for how ever much beyond 0.2 you can get in there. Having ground come loose / bounce is not a good thing.
I'm still not quite seeing how you are getting all the ground and power duplicated on both sides. The pcb layout sure looks like some of the ground is under some of the power.
I'm still not quite seeing how you are getting all the ground and power duplicated on both sides. The pcb layout sure looks like some of the ground is under some of the power.