[2021] Can't Play w a Friend, slight version difference

FYI DLCs do not affect checksums at all. But any mods in use must be identical of course, even their load order may need to be the same.  

The launcher CAN preserve mod load order, but I believe that is implemented in some games and not others, according to the wishes of the developers of each game.

Anyway you do have a mod issue if you can play MP together unmodded. Try both enabling just one mod, test, enable another, test - you should be able to pin down which one is the problem that way.  

What if you both opt into the stellaris_beta version, in the Steam Betas tab for this game? That uses Paradox MP services instead of Steam's.  

stellaris_beta ought to be identical to the current Steam version, except for using Paradox MP instead of Steam.