The ship needs to be iconic. Examples from other games: Sapphire yacht in Transcendence. Endless Sky may or may not have a mascot, but the Berserker (an interceptor one step above the starter Sparrow) is the app icon.
For id FPS, probably either Cacodemon or Cyberdemon for classic Doom. Likely Shambler for Quake.
Nobody wants to pilot a weak ship appropriate for a helpless civilian or villain mook. That rules out stuff like shuttles or Hound.
If you want a relatively early ship that is somewhat heroic, well-balanced, and iconic of Starsector, Hammerhead is the best fit. If you want a frigate instead of a destroyer, maybe Wolf since that is classic, somewhat common, and is one of the starters. If you want something rugged and more civilian, Wayfarer could do it despite being a relative newcomer.
If you want powerful bad boy with an attitude, like Darth Vader from Star Wars, Cyberdemon from Doom, Shambler from Quake, or Ur-Quan dreadnought from Star Control, as well as iconic of Starsector, Onslaught is the best fit.
If you need a second bad boy to go head-to-head with Onslaught, like Freddy vs. Jason, Paragon is probably its best fit rival.