"We would like 2" to 3" thick disks that we could screw and glue together. "

Use more disks of thinner material.

Hardwoods are normally sold in quarter inch increments of thickness, specified as 4/4 (1 inch), 6/4 (1.5 inch), 8/4 92 inch), and even 12/4 (3 inch).
The price per board foot goes up with thicker sections (along with the number of board feet they contain).

Thank you Brickeyee, now I know what to look for as to size. What wood do you suggest for the centers of the bar stools?

"What wood do you suggest for the centers of the bar stools?"

Any good hardwood.

A lot depends on what color you want and how the filler pieces will be stained and finished.

Unless you take steps a lot of end grain will be present in the edges of the disks and will need careful attention to avoid having it colored very dark by stain or dye.

To increase the height of swivel bar stools, simply buy 6-10" furniture legs and turn the tops to match the stool legs. Drill the size whole in the middle to allow the legs to fit snugly. Stain to match the wood of the stool. Glue leg if needed. Very quick and simple way to increase height without increasing the distance from the seat to the footrest.

I have a tropical style barstool that needs to be raised it has three bamboo legs how would I raise that