I know the idea of a despawn timer would be nice with numbers, but I could imagine that would be too taxing on the system.
So I suggest a simple solution that involves adjusting a graphic that is already there.
So first, don’t worry about the commons.
That leaves us with rares, epics, and unique.
Use the color circle rings of those dinosaurs.
With ten minutes left. Have the color circle rings start to flicker once every 5 seconds.
With two minutes left, have the color circle rings flicker once a second. This creates a sense of urgency and tells the player either to run to get in the desired shooting position or just shoot where you are now.
I think this one is more fun.
Shrink the color circle ring over time.
This is fun, because as a creature realizes they might be being hunted they get more wary, and plan their escape better. An experienced player would be able to tell by how big the color circle ring is, and adjust their effort accordingly. It is also a little more subtle.
Both methods can be made to prevent players to fully game the spawn times so they still appear random and should not be too hard to code or tax the system.