I don’t agree with the idea that there is increased liability of the designer here. The designer is never obligated to perform exhaustive enough inspections of all aspects of construction to find every bit of non-conforming work. Lots of things are drawn or specified that the designer never sees or specifically approves of. The obligation to construct per the documents falls squarely on the contractor. There may be a small liability if the designer creates a requirement in conflict with the manufacturer’s requirements, but there would be a lot of other factors that come into play for this to create a liability to the designer, principally negligence by the designer.
I’m still not convinced to include the MII as a submittal of any kind. Instead, Division 01, or individual sections, should just stipulate that the Contractor maintain access to MII on site, whether hard copy or web based. It is not necessary to place the information into the submittal stream in order to accomplish this. The design professional has duties toward informational submittals that are almost as rigorous as action submittals - the only difference is, a response is not required. I prefer to keep responsibility for compliance with the Contractor.