The Pet Treat and Lesser Pet Treat buff persists through death.
Be careful not to waste your treats. If the buff is already on you, using another treat won't reset the buff duration. Either wait for the duration to run out or remove the buff by right-clicking its icon before using another treat.
Battle Stone or pet treat
Re: Battle Stone or pet treat
Imo, definitely buy the treats. Battle stones are easy to get (2-3 per day from the pandaren tamer dailies), but you can only get a max of 2 of the 50% pet treats per week (celestial tournament and 10x pvp wins). That extra pet treat makes a huge difference in how fast you can level pets. On days where I only have the lesser treat and my Safari Hat, I can typically level 3 or 4 pets from 1-25 with the full tamer daily circuit. With the other pet treat stacked on top of that, I can level 5-6 in a day.