Do you insist on a breechloader? Or are you satisfied with rapid fire? And do you want rifling or not? The 3 are interrelated in their effects IMO.
If you'll accept a revolver-type weapon, a snap-in/-out cylinder (actually quite common for revolvers before the swing-out became standard) is a viable option; it need not have only 6 chambers, either. (Frex, the LeMat had 9.) You do have to overcome the Colt patent, but this can be done with cylinder mods. In a pinch, you can adopt a
turret gun.

(Probably not.

)
If you insist on breechloaders, it really demands metal cartridges. For those, the earliest AFAIK is the pinfire (patent 1812); if you've got an inventor developing a breechloader, it's possible he also develops his own *pinfire round (parallel OTL Smith & Wesson's rimfire). You could, just, maybe, get by with cardboard cases, but those are so suceptible to weather....
Rapid-fire smoothbore is one approach. Slower-firing rifles, with an early *Minie ball, is another way of achieving much the same effect, because it exposes the enemy infantry to fire more than twice as far away, effectively increasing the ROF. This has enormous & immediate impact on tactics & doctrine. (Compare Napoleonic Wars to, frex, ACW after Gettysburg. Or think of Pickett's Charge & realize the OTL standard issue Springfield rifles alone could stop it; just adding faster-loading conoidal slugs make that even easier.)
You've also got logistics to consider. Hi-ROF means hi ammo expenditure, & with horse-drawn transport, that's a big deal: more ammo means more wagons, which means more horses, which means more fodder, which means even more wagons... Before long, you've hit the limit of your ability to operate away from a rail line. (This really bit the Germans in OTL WW1.) This has very significant tactical & strategic implications. (If your enemy knows you can't operate more than x# mi from a rail line, or a river, all he has to do is control them nearby. Or cut them. {This may be asking for more sophistication than a 19hC general would have.

}) It was also in OTL AUS a reason (or an excuse

) to reject the Spencers & Henrys. It also changes the training regimen, from emphasis on aimed fire to emphasis on volume fire. (AUS was still training for aimed fire in WW2, when SL Garands had reduced the usefulness of & need for it...)
even if Prussian doctrine called for rate of fire over accuracy, soldiers still aim, otherwise it's a plan to get entire brigades' worth of fire to go over enemy heads repeatedly
An excellent point. One I overlooked...
...you basically negated the majority of the advantage of the Ferguson, while artillery will still kill you at three times that at least
Actually, it won't. In the ACW, the standard issue Springfield rifle musket could kill a man beyond the range of arty.