Checking back in to report our progress.
Note to others following our project: hydrated lime is a seasonal product (spring) at HD, but we found it left over at our local hardware store!
We managed to siphon down the pond to about 4" of muddy water. I got in there with a garden rake and managed to scoop out about 40 larger carp... pretty much all of the fish over 4" long.
We put some of them into our lower pond and most of them in a 100 gallon stock tank, with hose running in the bottom for oxygenation. Then, we limed the pond. SAD to see remaining small fish struggling, but it was all over pretty quickly. We broadcast lime on all slopes, and are now continuing to work on the swimming area barrier before the fall rains begin.