Trying to figure out how navigation is done, I was looking at BasicPopupMenuUI where I found two methods called next/previousEnabledChild(...)
that are checking the following UIManager property:
UIManager.getBoolean("MenuItem.disabledAreNavigable")
So after setting the LAF to windows I add the following line of code:
UIManager.put("MenuItem.disabledAreNavigable", Boolean.FALSE);
and it now works as you expect.