So, Animals caught in a Rabbit Trap are instantly despawned and the trap component uses a table of loot prefabs to spawn in a new prefab for you to acquire when you check the trap.
This means that traps will always give you a spider with full health and no loyalty, regardless of the original spider's state.
1. Capture injured or loyal spider with trap.
2. Pick up the trap as Webber.
3. Notice the spider that's now in your inventory lacks loyalty and/or has been magically restored to full health.