We have resolved the issue using standard support. I am publishing the solution, in case someone else runs into the same issue.
Kentico does not fully initialize before tests are run, so the tests are executed before some parts of the application initialization are finished. To resolve this, you can add an init method to the test, to serve the same purpose as the CMSModuleLoader.Init() method:
public void Init()
SKUInfo.TYPEINFOSKU.OnLoadRelatedData += CMSModuleLoader.SystemObjectCustomizationLoaderAttribute.SKUInfo_OnLoadRelatedData;