I spent my mid childhood on an LCIII. One summer my friend brought his Performa over and we tried to play 1v1 Warcraft 2 over the serial port. LocalTalk or something alike? But it just never quite worked right. I remember how frustrated and confused my older brother was. The computers would sometimes see each other but would drop off so easily. Was this that?!
Iirc Warcraft 2 came out well after the LCIII was obsolete (in fact, I’d be surprised if it was ever released for 68k Macs). Official system requirements list PowerPC and MacOS 7.6 as minimum required[1]. Maybe thinking of a different game? [1] https://www.blizzplanet.com/blog/comments/warcraft_ii_tides_...
You can just about make out "68040" in the requirements on the box art here https://images-worker.bonanzastatic.com/afu/images/7569/5203...
Edit: found a photo of the expansion system requirements, also lists 68040 as supported https://cdn.mobygames.com/covers/4056502-warcraft-ii-beyond-...