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Date & Time: 6/28/19
Frequency: Ongoing
World name: Coeurl
Character name: Scox Littletoes
Retainer name: n/a
NPC name: n/a
Monster name: n/a
Class/Level: Scholar/77
Party or solo: Both
In-game time: ongoing
Area and coordinates: n/a
Housing: n/a
Steps:
1. Go into Character Configuration
2. Select Display name setting
3. Display name settings for Pets, Change to "Always" or "During Battle"
4. Engage in battle and your Pets nameplate will disappear, it should stay visible

I've tried to change settings from "Always" to "During Battle" both settings yield the same results, my Pet's nameplate disappears during battle.