In my opinion it's not a big deal to use a GitHub. GitLab just seems nicer, ...and is open source, ... and free, ... and is not bought by some tech gigant recently, ... and has no travis as a CI tool. You know all the reasons.
However - it's not a problem to use two or more remote repositories. It's a fundamental feature of git. More info: https://gist.github.com/rvl/c3f156e117e22a25f242
Mirroring to github can be done automatically as a one of gitlab's CI's pipeline stages. So that the whole project's life can be held on gitlab.