Years ago, in the early 90's maybe, there was an over/under side hammer rifle, put out by Ithaca, seems to me.
The only write-up on it I saw, noted ignition lagging or misfires for the bottom barrel due, I suspect, to the long flash channel from the powder to the cap. Of course, if one loaded the rifle with that in mind, a thump on the ground or with the hand would introduce powder into that channel and solve the ignition problem.

Nice rifle, TH44. Looks like a bayonet mount for sure. Typical "European" design. Seems to me the Baker rifle and the cap-lock Swedish Kongsberg military rifles had very similar bayonet mounts.