Closed cooler = bad idea, as you said.
Either drill a few holes or find a container that has a few holes in it or that you can put a few holes in. Airtight is not good.
A pillowcase and small a container that you can keep under the covers with you - again - not airtight. You just want to make sure you don't roll over and crush the BP - hence the container.
Hope things improve quickly.