Do you know if GitHub Enterprise imposes an API limit for unauthenticated requests? If it does, does it support personal access token? (See https://github.com/buunguyen/octotree#github-api-rate-limit.)
Depending on the answers, it could be very simple change, i.e. allow users to configure API subdomain. Unfortunately, I don't have any GitHub Enterprise deployment in handy at this moment to find out.