No, the reason it's not a Bootstrap Paradox is because it isn't a paradox.

Example of a paradox:

I go back in time. I kill my father. I am than not born. Which means I can't go back in time. And I can't kill my father. So, I'm born. And this repeats.

"At the most basic level, a paradox is a statement that is self contradictory because it often contains two statements that are both true, but in general, cannot both be true at the same time."

There is nothing self-contradictory about the Chrono lives situation. There is no conflict with itself, no self-contradiction. Ergo, it is not a paradox.

A bootstrap paradox DOES contain self-contradictions.

To use the example given by Dr. Who: "A huge Beethoven fan goes back in time to meet his idol, only to find no evidence that he ever existed. Shocked that the world would miss out on his musical genius, the fan takes Beethoven's place, publishing every piece of music and assuming the role, while history keeps going with barely a ripple."

The paradox here is that the fan got his music from the past... But the music in the past came from the future. This is what a bootstrap paradox is. It's a stable paradox because the information exists in any case, but it's a paradox because it has no origin. It just is.

In the Chrono situation, this conflict is not there. Chrono, the doll, the knowledge of using the doll to replace Chrono... All of these have very clear, linear starts and ends. None of them are infinitely repeating upon themselves, and therefore, none of them are a Bootstrap Paradox.