There is merit to this idea, especially if Gwenn lingers in pretty places like Harper’s Ferry WV or Charleston SC. The issue is reliability of the fridge at the stops. The insulin is the driving need. My source was
36F to 46F is not that hard to maintain with ice in a cooler. The butter will be fine. The key is knowing what the temp in there is. A remote thermometer would be great. Any thermometer is fine as long as you remember to check it every time you look. Without other information I’d shoot for staying in the range of 36F and 40F for some margin for warming up, but watch carefully for getting too cold. Apparently freezing insulin is bad. I suspect without knowing that the 36F to 46F already has some margin but we don’t know if that’s true and if true we don’t know how big the margin is. I’d still talk to a trusted pharmacist.
Remember (thermodynamics, heat transfer) that there is a state change. A lot of energy needs to come out to get from liquid to solid. This means liquid at freezing temp is different from solid at freezing temp. Check with that pharmacist about what the freezing temp of insulin is. It probably is not 32F (organic chemistry).
Don’t leave the cooler in a hot car during the day if you stop for a day. @MsBean’s idea of using a fridge at your stops does have merit. HOWEVER, I would not do that in a hotel. Aside from not being sure about temperature, I’ve moved mini-bar/snacks out of a hotel fridge and had housekeeping put all my stuff out on the floor and reload their stuff inside. Better I think to leave your cooler on the floor of the room in air conditioning and leave the mini-bar fridge alone. Hotel ice machines are usually pretty good.
