If I recall correctly, you are now past 100 days of processing?
In this instance, I would definitely pursue this with your MP. Typically, we don't recommend getting your MP involved because that route was abused so much in the past and MPs now do not have the same ability they once did to chase up a pending application - especially if you are still within the normal processing timeline. But if you are still waiting after 100+ days, and as you haven't gotten the "not-straightforward" email, certainly involve your MP at this point.
Hope you're able to get it sorted quickly.