If I decide to perform rapid mobilization at the beginning of command phase and I choose a new location. Do I immediately reveal the old base location or do I just hold onto the probe card I selected until the end of command phase? Thanks for the help in advance

You reveal and resolve the card immediately. Then it sits there until the end of the command phase. Then you start to resolve it. You don't even pick option A or B on the card until that time.

Heck the way it works, at the end of the command phase, you draw four probe cards, look at them and then decide if you want to move your base. Only if you move it do you reveal the slotted probe card.

You reveal the card, not the base, when you resolve the mission. You don't have to move the base unless you decide to after you've seen the probe cards, which is done at the very end of the command phase.

It appears that the timing was done this way to give the Imperial player one whole turn to capture the base if it was revealed but not captured last turn.