Technically speaking, the LAG bag is supposed to be alone. That being said, your experience will vary from airport to airport, some are more relaxed and tend to work with the passengers as long as the "dob" kit or toiletries bag is outside of the larger carry on bags. Some, not so much. As always, it will vary what type of response you get from the TSOs depending on the airport.

This is not really the way it should be, you should get essentially the same experience over the same subjects, but that does not appear to always be the case.