In your case, yes I think the MP can help you out.
Unfortunately, as of late, I think they are just trying desperately to churn through as many cases as they can. In that haste, they may overlook the "low hanging fruit" -- cases that only need a few minutes to review and approve. Additionally, it can be eMedical themselves, as they are not affiliated with CIC, and does work with other governments to provide front-line approvals on medicals as like a "second set of eyes" on the details submitted by the physician.
An MP can make a call on "the MP line" (that's an actual thing if you order your case notes, you might see it), and it might provoke a change. One such instance I recall was a DM local to me, who was waiting forever for an office that, two weeks to a month earlier had given us a landing appointment within 2 hours of ECAS saying DM. Once the MP's office actually became engaged and called, there was no reply but a landing appointment came that morning.