I pumped after a feed and left it out at room temp for the next feed. Any extra pumps I popped in the fridge.
I would pump before I went out and take the fresh milk with me, that way no heating required

I agree you should defo keep offering the boob. And lots of skin to skin time. We have gone from no attachment and full time ebm to exclusive BFing. I know it's harder with the extra kids but feeding ebm and expressing takes up so much time.
Good luck!