I don’t feel like there is enough room inside the case to limit the noise. Most of it comes from the second stage piston pump (see video) and the noise also escapes through the air inlet which you can’t really block. In terms of heat, I believe these pumps don’t really have an issue with the additional heat but the case does get almost too hot to touch (especially closer to the air connectors).

A better approach might be to build a “dog-house” type of insulated box (one side open) that you place over the pump and direct that opening away from whatever is sensitive to the noise. Do share what you build!