Hi,
Yes to your question about the two plugins potentially messing with each other if not configured right.
One thing I can tell you for sure is that dynamic caching by Cloudflare definitely causes known issues with LSCWP. The way I see it, your best shot at troubleshooting this issue is by process of elimination. First, disable Cloudflare caching and see if the problem goes away. If so, then you have two options. 1) Not to user Cloudflare caching anymore; 2) Adjust settings in Cloudflare until the problem is gone.
If, however, the problem persists even when Cloudflare caching is disabled, then we’ll need to look at the LSCWP settings and see which of them might be the cause of the problem. Off the top of my head, the only thing I can suspect is the “Serve Stale” setting under the Purge tab. I see from your report that it is enabled.
Regards.