I just unlocked he Lazuli (Yay!) and a new medal quest appeared: "Our Former Oppressors." To complete the quest, you have to disable 50 Lazuli Harrower ships. Now, I have never seen a Lazuli Harrower ship in my NPC battle tab (I currently have 6 slots open), so how often do they show up? I haven't changed my race to Lazuli yet, but if I do, will that increase the probability thatf the Harrower ships will show up in the battle tab?

It's a matched NPC, so you should be able to see it, the RNG will summon one eventually

Uncommon Npc, quite rare to see as I was told from other legionmates that had finished the medal. They said that it was annoyingly long to finish. Grats on unlocking Laz. Become a Physicist for double the base buff.

Becoming a Lazuli Fixer would be a better choice, no? Doesn't give a double base buff, but it does give a greater benefit.