Caddy is great for web servers, but it's still not possible to have it run commands post certificate provisioning. So it's kind of a non-starter for anything but web-servers as there is no way to tell a different system to reload certs.
We're working on this! Hoping to have our new event dispatching system ready in the next few months. This'll let you hook into the post-issuance event and do whatever you want afterwards.