Vector layer is blurry if it's in composition
About the only thing to do is scale up the nested comp (the one with the puppet animation in it), so that it matches the size it needs to be in the outer comp without having to zoom it.

There's a script that will do this for you, if you look at the File>Scripts> menu there should be one called scale composition.

This script basically just attaches everything in the comp to a null, resizes the comp using the comp settings, then scales the null to match. So if the script doesn't work for you you can do it manumatically.