six-: Check if the folder for saves have somehow become write protected.
Not had time to play yet myself so no idea if I'd have the same issue.
Just gave it a try and saving/loading works fine for me on updated game trough galaxy.
Definitely wasn't the save files being write-protected on my side, I was shunting a lot of files around when I was trying to resolve the issue so was checking them for that sort of thing whilst I was doing that (I take a lot of save file backups, paranoia given to me by a certain Fallout New Vegas early save corruption bug). Can't speak for others though of course.
The fact several people seem to get it simultaneously, and mine started doing it after working absolutely fine, suggests external influence. People saying the GOG site was screwy earlier today, combined with my experiment with the communication exe, does point to the GOG Galaxy connection being the trigger, although the root cause is probably problems in both the GOG Galaxy thing and Battletech itself (a game really shouldn't fall over whilst saving locally just because an entirely unrelated external server is throwing a bit of a wobbler).
Also probably means that once those GOG servers are working again, people who were having this problem should be alright again, it was a fair few hours ago so hopefully everything's been sorted for now and fingers cross you shouldn't see this (well, until the GOG servers fall over again).