Federation Commander

When a ship is in an asteroid hex and fires a seeking weapon out of that hex, does the seeking weapon take damage from the asteroids in the starting hex?

We could find rules for seeking weapons traveling through an asteroid hex (takes damage) and for impacting a ship in an asteroid hex (impacts before asteroid damage), but couldn't locate the rules for launching out of an asteroid hex.

No, it does not take damage.
While there is no "actually" wording spelling this out, the key is in the example:

Example: A drone enters a hex...

That is correct. The key point is entering the hex, not exiting the hex.