There's one way to find out, Bob.
For an avowed M&P fan (such as yourself) to have not even tried it is - to me - unbelievable.
I replaced all my innards for my M&P Pro with Apex parts to improve trigger etc. I felt good about the move but my scores did not reflect the lighter and shorter trigger pull. My Pro had the 1 - 10 twist and shot better than I could so I am pretty sure the barrel would not have improved my scoring significantly either.
I have a Pro 40 with all the Apex goodies including their FF trigger. It has developed a bad case of the light strikes. I have replaced the striker and will do some more testing before contacting Apex to seeif they can help.
If the Apex barrel is actually fitted by a 'smith there might be a slight gain in accuracy. If there is and the shooter is good enough to notice it is all good. My experience has been spending money on some decent instruction and lots of careful practice will improve results more than a new barrel. The M&P's are pretty good pistols.
I have a stable of PPQ's. Out of the box they have better triggers, shorter resets and are capable of better mechanical accuracy than the M&P's and are, like the M&P's dead nuts reliable. I think I still shoot the M&P Pro better but that likely is because I have more time on the gun. So there you go.
If a shooter thinks he/she shoots better with their blaster by adding parts so be it. There money, free country. I just have to say though from my experience there are few shooters that I know who can take advantage to any great extent a shorter reset, a slightly lighter trigger pull etc. Short term, maybe a slight improvement but really over time...not so much.
Now practice, not just shooting holes at paper but practice and some instruction and you see improvement. Even then we are all not like Robbie Leatham.
Take Care
Bob
PS I shot a WA State Match in Revolver Division with the, then current IPSC World Champion. Nice fellow who helped this old Canuck along. My take away. If I practiced the rest of my life, had the lightest, fastest revolver ever made I could not hope to shoot near his level. It was not what he did shooting that allowed him to wax my butt but a hundred little things that separates average results from winning results.
To the OP spend $300 on ammo AMD decent instruction and your shooting will improve....
Take Care
Bob