I don’t know if Weaver II supports anything called Cornet Cache. If that is the successor to Zen Cache, it would depend if the new version changed internal function names or not. There will be no future changes to Weaver II to possibly support other plugins.
If caching is important to you (it usually only helps a little bit on very busy sites), then you really have 2 choices.
1. Disable Smart Mobile mode which makes the output completely responsive, so any cache plugin will work.
2. Switch to Weaver Xtreme as it works with any cache plugin.