I used to pump under my nursing cover at my desk during my lunch. I also bought a cooler bag from Vcool and some ice packs / ice bricks as well. Top part can keep my Lacte pump, charging cable, pump parts and a bottle. Below part is the cooler compartment where I keep the ice packs/ice bricks. There is also a small compartment so I brought a small bottle of bottle cleanser and small bottle brush to wash after use. I have to leave home at 7am and will only be back at 7pm. It can last me till then. Once reached home, I’ll quickly put in the freezer.
Can pump under nursing covers, maybe let your colleagues know before hand that you will be pumping. Over the time, they should get pretty used to it. For storage wise, you can look more into the “Fridge to go”. Depending on what time you’re pumping, be sure to look out for those that can last longer hours. :)