Was having this issue too. Try updating the wrapper to use a different version of Wine. I tried version "WS9Wine1.7.15NoFlicker" and was able to successfully complete that level as the Dragon clan without crashing.
For those unfamiliar with Wine, you can do the above using
Winery):
Step 1: Launch Winery; click the "+" and select the appropriate engine to download ("WS9Wine1.7.15NoFlicker" in this case) (you may need to first refresh the list of available engines using the circle-arrow button). Once the engine is installed, quit Winery.
Step 2: Right-click on Battle Realms.app, choose "Show Package Contents" and launch Wineskin.app. Click on "Tools" (you should now see displayed, near the bottom of the window, the engine the wineskin currently uses). Click on "Change Engine Used" (under "Wrapper Tools") and select the appropriate engine ("WS9Wine1.7.15NoFlicker", which is now available thanks to the previous step). Click "OK" and wait for the engine to install, then quit Wineskin. You should now be able to run Battle Realms.app with the appropriate engine.
General note (applies to any game run using Wine): the latest engine version isn't necessarily always the one you want (e.g. the engine version I list above is not the latest one available, but latest ones didn't work around this bug). Experiment to see what works best for you (fortunately Winery makes this reasonably straightforward).