Side note: the Batch Project tool has been problematic in the past, so I'm glad it's working for you. You can also right-click the Project Tool and get a batch version of it.
I don't know why you're getting the same (good) results whether or not you set a transformation in the Batch Project tool. It could be picking up the transformation from the an internal active list of transformations (GeoTransformation Operation Set).
We don't set any transformations in ArcMap, you need to set it to match the Project tool, as you've found.
Melita