I don't know about the pump bag. I did have just one tote bag and one small cooler I carried on.
Frozen milk requires no extra screening.
For liquid milk, I just had bags and bottles that I put in a small cooler and filled with regular ice. It is possible the TSA agents will give you crap about the ice, especially if it is melty. I did not have any problems with United in Austin or SW in San Diego.
Are you traveling with or without your baby? If you are going more than a couple of days without your baby, milk builds up fast and you may want to check a large cooler with the milk. You could use bungie cords around it to make sure it stays shut.
And an FYI, most airports do not have a good place to pump. In Albequerque the family bathroom had power outlets but no chair. I stood and hand pumped and charged my phone, ha! The Houston terminal I was in had the family bathrooms occupied and I sat in the main hallway on the floor with a cover.
Good situation at the airport--look for an empty gate. It will be out of the way. I did that in San Diego and there were even plugs below the seat.