Unfortunately the linux installers for Battletech (battletech_1_9_1_686r_36568.sh) obviously contain some older/incompatible libraries for some linux distributions. On my Fedora 42 PC the game was installed without any visible problems, but when I tried to start it from the menu I just got a black screen after the initial few screens showing the Paradox logo, etc.

Next I tried to start the game in a terminal session from the command line: (replace username with your user name) I went to the games' directory with
cd "/home/username/GOG Games/BATTLETECH"
and typed
./start.sh

This produced these messages during startup
Unable to preload the following plugins:
libBink2x64.so
libc.so.6
libopenal.so.1
libsndio.so.6.1
and therefore - after locating some of these libraries on my system - I went to this directory
cd "/home/username/GOG Games/BATTLETECH/game/BattleTech_Data/Plugins/x86_64"
and moved 2 files up one directory with
mv libopenal.so.1 ..
mv libc.so.6 ..
and created new links to Fedora's files in /usr/lib64 with
ln -s /usr/lib64/libopenal.so.1.24.2 libopenal.so.1
ln -s /usr/lib64/libc.so.6 libc.so.6

After that I returned to the games's install directory with
cd "/home/username/GOG Games/BATTLETECH"
and started the game with
./start.sh

Now the game still complained about 2 files with this message
Unable to preload the following plugins:
libopenal.so.1
libsndio.so.6.1
... but appart from that everything runs fine now and I have already played the first two missions without any issues.

Maybe this information helps some other linux users to run Battletech on their systems - good luck and have fun!