I bought the Planetary Pack a while ago and when I launch it, I get the original Alpha Centari, but I can't get Alien Crossfire to appear. I've reinstalled it and didn't get an option for it.

There should be two executables in your installation folder; terran.exe opens vanilla SMAC, and terranx.exe allows you to play Alien Crossfire.

If you prefer to use Galaxy, instead of "Play", click on the options button next to it (called "More" in older versions of Galaxy), then on the displayed menu click "Additional executables" and select the expansion (Galaxy, by default, launches the vanilla game).

In GOG Galaxy launcher, click on the 'options' icon to right of "Play".
Navigate to Manage installation > Configure.

Under "Custom executables/arguments", scroll to the bottom and click "Add another executable/arguments"

Under File 6, choose axstart.exe.
Set it as "Default executable".

Note that File 6 is your first user-made one as there are Files 1 to 5 already existing.

Click OK.

It will now run axstart.exe every time you click Play from GOG Galaxy launcher.

TIP: In file explorer go to game folder and right-click on axstart.exe. Pick Properties > Compatibility > "Run in 640 x 480 screen resolution".

This tip should fix issues from my default 1920 x 1080 resolution which cut off the left and right-edge UI menus.