Hi Amber-
Thanks for the great question. The plugin you’re seeing is a fork that A2 Hosting created from W3 Total Cache, but it doesn’t look like it’s listed on their GitHub account any more, so it’s possible that it is not being maintained any longer.
I’m not totally sure what changes A2 made for their version of W3 Total Cache, but I haven’t heard of any issues running the current official version of W3 Total Cache on A2’s systems either.
To be safest, it’s probably best to reach out to A2 themselves to see what their recommended upgrade path is. It looks like the current version of A2 Optimized WP includes its own caching, so it may no longer be recommended to run them both together like it used to be. At the very least, I’d recommend making any major changes in a staging environment rather than your live site.