With the problems people here are having trying to source Blue Cats, I think you are much better off stocking the Hybrid Striped Bass and more Largemouth Bass along with the Tiger Muskies.

Since it's an old sandpit, how much spawning habitat is there for the Bass and Bluegills?

The easiest way to help control the numbers of rough fish and fish that you don't want to have in there would be a combination of gill nets and hoop/fyke nets. The gill nets could be run 2x daily and you'd get a LOT of carp/gizzard shad out, and you could turn loose any other fish that you didn't want to remove. The hoop/fyke nets do a great job on catfish.

The HSB would do well in there because they'd be occupying the same habitat as the Gizzard Shad. They aren't that expensive to stock, you could stock a bunch of 4" ones and they would grow fast. If you were here, it'd cost around $25-$30 to stock 10 Hybrid Striped Bass/acre.

If you wanted to target the carp and grow the BG bigger, put a Texas Hunter Fish Feeder on the dock and feed the fish. The Carp will get accustomed to the food and you can either bow fish them out or catch them on rod and reel. The Bluegill in the lake will also find the feeder and start feeding. The Largemouth (and HSB if you stock them) will also find the feeder. The HSB will eat some of the food, the LMB will eat the smaller fish that are coming to eat the food.