I'm not too clear on the use of a target designator for something like a missile or mortar.

As I read it, it seems to say that if you're using a target designator, you have to make a ranged attack with the designator, using something like the Pistols skill.

My character uses a mortar mounted on a drone such that it fires and reloads automatically when I use the target designator. Basically, I highlight the target with the designator, signal the drone, it fires on the mark.

What skill would that work with?

Our GM has ruled that it's a Pistols skill roll with the designator.

Pistol skill, needing 1 net hit sounds about right.

You're basically doing a ranged-touch attack, to borrow from the game that causes cancer.

The details are in Arsenal.

Sadly, target designators are exotic, last i checked. However, Smartlinks are capable of remotely locking onto targets and guiding munitions onto them. However, they do so at a bigger penalty than target designators

See Information Guided Warfare. Also Tactical AR Networks have this capability. Indirect Fire rules as well.

If you don't wanna break out arsenal's expanded/optional rules, though, it seems your gm's got a houserule for it.