It's not possible currently. What you can do is duplicate your current dashboard and have one with only local access and the other with only cloud access. This way you can edit the URLs to whatever you want.
Yes, it's not ideal to have 2 dashboard links but that's the only way for now.
You should use mysite.com for both links and use dns in your local network to provide the right response in the right context. Depends on what your router is but some can do this and others you need to setup a dns server locally and point your router to it