I found an old thread here with users doing similar testing and running into issues because of the basic auth prompting in Live Links. The TLDR is that it seems like these sorts of things are better tested on a live site.
One user came up with a pretty hacky workaround here but this might be outside of your comfort level to try:
It seems like Jetpack testing requires a site to be live as well. They have some steps on how to set up “Offline Mode” but a lot of features will still not be available: Offline Mode