When I ran my SIPs I went without any mulch or cover - but also occasionally would top-feed/water in some extra goodness for a nutes/microbes bump. It worked for me fine, including my best run ever. Now, keep in mind I was in a 4x4 space but that was in an open room around 5X10 so the humidity contribution wasn't a factor.
I have purchased some EPDM rubber pond liner to make covers for the next time I run the SIPs and/or I'll use the original tote lids and leave a small air gap between lid and top of soil. I'm not sure if the pond liner would be able to be kept lifted enough to ensure a small air gap exists which is why I may fashion something out of the lid instead or use the lid as a support system and have it cut out enough so the pond liner is really just providing a better seal around what opening I make & the plant/it's stem.