I don't see how a higher flow would cause the baskets to pop up, since the water is coming in at the top and out through the bottom. Does yours not have an adjustment in the bottom to moderate the flow rate?

A picture would help a lot. Does this pool have a floor drain also?

Try putting a small rock in the skimmer basket to weight it down so it doean't pop up every time the pool pump starts up.

I used to keep the metal diverter that attaches to the vacuum hose and plugs into the skimmer drain in the basket to keep the basket from moving...and so I wouldn't lose the diverter.