Minecraft offline

I would disconnect your lan from the internet, thereby allowing you to play offline, then open your game to lan as normal.

An easy option is to edit you hosts file (Windows and Ubuntu at least) and add:

authenticate.mojang.com   127.0.0.1 #or anything else

So every attempt to login the Minecraft will receive a null response then enable the Play offline option.

Your computer is trying to connect to mojang servers, but is failing, and is stuck in a retry loop. You should disconnect from the internet, start minecraft in offline mode, and then reconnect to the internet.

There are other to accomplish this, but they're all more complicated than you need. You could blacklist the mojang servers in your lan router, or your HOSTS file, like Aragón's suggested, but that only makes sense if you absolutely have to use other internet programs while minecraft is still online. (or planning on not having a valid connection to the mojang authentication servers, which there is no legitimate reason to do, but....)