Ok I've gotten it to work. This is what I did:
1. First follow the thread shared by teceem and the GOG suggestion for fix. Basically open wineskin, by going to LOTR2 icon in applications, right click show package contents, Contents, Resources, game, right click again LOTR2 icon show package contents, double click Wineskin. Then set screen options, uncheck auto detect GPU..
This now allows you to launch the game, however it quits after a few minutes of play.
2. In order to be able to play it, I had to change my engine. For me, I checked the engine used for another GOG game I have from the 90's (Total Annihilation, which works fine). That Engine is WS9Wine2.22.
You can change your engine in two ways. First by going back to Wineskin as described above, then click Advanced, Tools, Change Engine Used. If you have another enigne downloaded try it out. If you do not have another engine option to choose from, you must download Wineskin from your browser again, and during your setup pick another engine, try the one I used.
This worked for me, so try it out!