My Uberti 1851 London Navies will consistently give me a quarter-sized group at 15 yards with 20 grains of Schuetzen and a round lead .380 ball.
This pair (cowboy action shooter!) happened to shoot to point of aim out of the box, but all the other 1851s and 1860s I have owned in the past shot way high, and I had to grind and file the rear sight notch in the hammer to bring them to point of aim. I got lucky, but this is the exception.
Piettas usually have good arbor fitting, but you should check it anyway. If the arbor does not fit the barrel assembly properly, accuracy will eventually suffer. Piettas usually need some additional fitting of the bolt into the cylinder notches. If this is off, you will see burrs and wear on the notches and lead-ins after a few cocking cycles. Reliability will also be enhanced with better nipples.
So, yes you could tune an 1851 to be a 15-yard small game gun, but it's unlikely to be as good as you want out of the factory box.
You can find links to Pettifogger's tuning procedures at The Open Range forum, or someone more savvy than me can probably post a link to find them quickly.