That worked!! I really wish I could buy you a beer
I think one of the main trick is to not let OpenXCom access the "default linux" data folder, but instead force it to access the data folder that's specific to the new installation.
Using your method, when I ran just "$HOME/bin/OpenXcom/bin/openxcom", I got the same crashes as I used to have before: As soon as your load a game, start a campaign or start a new battle.
When I run it while specifying the data folder, it works flawlessly. This seems to indicate that bslaveboy was right saying that we were missing a file. It's not really a bug (since it actually does come in the installation), but it's "feature" of looking first in the default folder, which for me was full of stuff from 1.0 and missing other needed things.
Edit: So I just realized that the -data option seems to add to the search path for data, it doesn't overwrite the other ones. Indeed, my "$HOME/bin/OpenXcom/share/openxcom/data" folder is pretty much empty.
One of the unfortunate side effects is that the game doesn't use up to date language files, so I get "STR_BLA_BLA" in game sometimes. It happens since the game finds the language file in the default directory: "/usr/share/openxcom/data/", but that one I got with 1.0, for which features like menus in manufactoring were absent (so I see STR_ALL_ITEMS instead of "All Items").
One fix, I guess, is to remove the offending folders from the default path and only keep things like mods there, while making sure each installation has all it needs in its local data folder.