Thank you all for the great ideas. I don't really have any interest in adding yet another bag to the rig to be frank. As much as I love accessories for the bike, I don't want to fred up the retro sporty look to the bike. I did manage to include the tube though. I undid the straps that hold the bag to the seat rails, lowered the bag, placed the tube between the top of the bag and bottom of the seat and then re-attached the straps around the seat rails. It doesn't look horrible but I'll explore other options should this not work long term.

Thankfully, the seat bag sees important use on weekend fitness rides when I ditch the panniers. Monday through Friday, I've got a change of clothes, lunch, extra crap, and my lock in my panniers.