A lot of gunsmiths, including the one I call my own, won't tune a trigger below about 4# for a "customer".
Liability is the big issue. Period. If pressed I know for a fact the number one response will be that triggers that are below that magic 4# number are not "street ready" because they are too sensitive and could result in a ND.
Now if you know a gunsmith that knows how you handle a weapon, and trusts you, they will tune it down below that number on a "carry" weapon.
Having shot the MP previously I will say that the trigger sucks as factory issued. Period. No middle road, no "what if", it's just not very good.
Having it replaced with a complete competition trigger is NOT a good idea in my opinion IF the weight of that trigger drops into the "one pound and ounces" category, which is where some competition guns reside.
Personally? I would look at having the weapon's trigger tuned by a local, reliable, competent gunsmith. Ask him/her to "reduce the take up, give you a clean, crisp break and reduce the over-travel to a bare minimum" and see what they can do for you.
A trigger job can run from $35 to $70 depending on the shop and the arrogance of the gunsmith. No offense to any gunsmiths here on FTF.