I don't feel birch is a suitable wood for cores. If you want something you can turn round and will not move and use quickly and is on the lighter side have a look at hickory. You can get dry 5/4 boards most any hardwood store and it's not expensive. My experience with it is that unlike a lot of woods that seem to hold tension internally hickory due to it's grain pattern seems to be dead straight and stay that way. You can tell when you saw it into squares there is no movement on the kerf as you progress along the board. Then when you turn it round it stays dead straight forever more. Maybe I was just lucky with the boards I bought but I love it for light weight cores. The porous nature of the grain makes it hold glue very well for cores as well. It will look a bit torn out when turned but not deeply. A quick pass with 220 grit sand paper and it's ready to go.
Edit: Never ever choose something as important as the core of your cues due to convenience.