So if I split the 12v supply between the LED strip and the servo at the same time, will the LED's brightness not be affected?
No it won't. We are talking about a regulated supply. The voltage will always be 12V up to the maximum current of the power supply. If you do not have enough current then get a bigger supply. You can also use a switching regulator on the servo supply, that will make the current draw less as they are about 90% efficent.