Try pulling the package sidekiq-pro which depends on sidekq from a virtual repo containing both repos.
After testing, I was able to get the sidekiq-pro package directly from a remote repo pointed at https://gems.contribsys.com but it fails if I try to get it from a virtual containing the remote repo.
Workaround is to have two sources in the .gemrc file but this does appear broken for virtual repositories.
1. Create remote gems repository pointed at https://gems.contribsys.com using our credentials
2. Create a remote repository pointed at http://rubygems.org
3. Create a virtual repository containing both remotes.
4. Change your .gemrc file to the virtual repository
5. sudo gem install sidekiq-pro
6. Change .gemrc file to point to the remote directly