The first question is whether you actually need the shorter barrel. If you find it to be a good fit for what you do, then we are on to the next consideration.
I have test fired a UMP in .45 caliber. It has very excessive recoil and stock flex, and the weapon porpoised like you wouldn't believe. When you consider the world's worst trigger, it was an all around flop.
I consider short barreled carbines to be supremely useful. The 10.5 will have all of the terminal effectiveness that law enforcement requires. Overall, it will do a lot more damage to a person than a pistol caliber will, regardless of the platform that delivers the bullet.
A word on ammunition selection. Our ammunition forum is full of advice, but for the short barrels, the best advice is to get Hornady 75 grain TAP, in the NATO pressure. Good stuff.
If you wish to go with a 10.5, or an 11.5 for that matter, you have lots of options, of which a few are good. Colt makes 10, 10.5 and 11.5 inch barrels, available from Specialized Armament (SAW)in Chandler, and LMT sells 10.5s. The best source for LMT uppers is Wes Grant at MSTN, available here on the Industry board. If you go MSTN, you will need to aquire lowers from some one else or use what you have.
The downside to a short barrel is that the timing will often be off. Although my own short barreled experiences have been trouble free, many people see the need to install a heavier buffer to delay unlocking and a stronger extractor spring to ensure reliable extraction. SAW and MSTN both see to this in their builds.
Lastly, John Noveske makes an excellent 12.5 inch barrel that has roughly the same muzzle velocity as the M4 that you are using now, but is shorter, more accurate and retains accuracy longer. Retail on these is 330 dollars. They are not for the budget minded, but they are the best. They are available from Wes at MSTN, assembled into uppers.
There are a lot of brands and other issues that I see fit to omit at three in the morning, but if you have any questions drop me a line through site IM.