The driver used on the Bobcat servo board is a constant current driver, so you don't use individual series resistors. R7 is used to set the current drive for each LED. It's not like the traditional case of a resistor per LED connected to a constant voltage.
The currents spec'ed for that RGB LED are likely maximum currents; I would try driving it at 20mA on each color and see how it looks. Your calculations look right to me; 1k would give 18.75 mA per channel. Unfortunately, the way the driver chip is designed, you get the same current on each channel.
/mike