Speed your 'play' problem is not unusual as even original S&W top breaks had this problem. So how did the S&W factory solve this problem?
I've spent a lot of time with original No.2 frame SA's and DA's. More parts than you would imagine are interchangeable even with productions dates 40 years apart. The factory adjustment was the shaft diameter of the barrel latch screw. I have measured no less than 4 different shaft diameters on the No. 2 frame top breaks that I have. That difference is the adjustment between the barrel latch/rear sight and the barrel latch lugs on the frame. Smaller shaft diameters make the lockup looser and the larger shaft diameters make the lockup tighter. I have no idea if Uberti offers latch screws with different shaft diameters to 'tune' the lock-up of any of the Uberti made top breaks.
The posted pic is of a pair of Second Model 38 SA framed (dated to 1881 and 1883) revolvers with Third Model 38 DA 5" barrels and Fourth Model 38 DA cylinders (post 1900). The better metallurgy in the post 1900 cylinders makes the light smokeless powder loads I create much safer to shoot. Light loading is to not abuse and be more gentle on the BP frames