Nah, forested pets keep their names until they're found. The only time a forest pet's name will become available while it's still in the forest is if it's found, and the finder decides they don't want to adopt. Thus the entire pet is deleted.
Anyway. I always really liked the fact that you might get good names when searching the forest. What's the point of searching otherwise? Sure, there's color and species, but adding names in increases your chances of getting lucky.
It makes the forest more worthwhile. And it actually keeps things more fair. You say give others a chance, but with the forest, everyone's chances are completely equal. No factor of when you can get on, or whether or not you know the name is going to be lost soon.
It just takes patience. When the forest is opened up again, whatever pet you're waiting for will come out. If you keep your eye on it and ask to buy the pet immediately after it's found, you should have a pretty good chance of getting it.