I would agree, in fact, offering a letter from an MP may give the impression that some criteria are not being met, and encourage them to look into details more closely. An MPs letter would only really be of help if it had to go to appeal.
You don't need to present a strong case, you just need to meet the set criteria (at the moment).
Supply the required documents fully.
Our experince with the embassy was good. They were polite, open and helpful, and went out of their way to help.
Of course this is based on the info you gave, maybe there is a reason you did not mention, but if that were the case, I think they would have expected you to not submit the application until you had everything you wanted to submit.