For both Star Wars: The Old Republic, and Elder Scrolls Online, installing them directly, or via Steam, causes the launcher to throw a similar error in either case stating that the process is out of virtual memory.
I know the computer is not totally out of virtual memory, because I'm on an M1 Max with 64 GB RAM. Plus, the system reserves some SSD space for swap, I think?
Anyway, this seems to be an issue with the emulated environment and the 32-bit heap the launchers run in (I think they're both 32-bit launchers). This happens when they're unpacking huge downloaded patch files, mostly.
The fix I found was to change the OS in winecfg to Windows 10. For some reason, this helps free up space in the 32-bit heap. It was enough to get these MMO launchers to finish patching.
I presume there's a lot of "noise" in the 32-bit stack that isn't there on native Windows due to all the translation layer stuff, and that's the reason it's hitting OOM. It only has 4 GB to play with, and they probably never encountered a situation where so much of the stack was consumed by the OS on native Windows, so they load the entire compressed file into RAM or something.
Anyway, this is not based on a deep debugging dive into the system, I just found that this step helped. Maybe it'll help Codeweavers figure out a patch so these launchers work out of the box, either through Steam or by installing the game directly in Crossover.