As you succeeded in firing an .38Special shell in it, your shot barrel must be chambered for a centerfire cartridge. The various Flobert numbers are all rimfires! A .38 Special case hit on the rim by aFlobert firing pin would never ignite! From your description I think your gun is chambered for the 9.1x40R cartridge. Though COTW lists it under "European rifle cartridges", it was often used as a brass shot cartridge in such garden guns. It's centerfire case was reloadable, so it was deemed cheaper to use than the 9mm Flobert. Here it is, from the 1910 Stukenbrok mail order catalog:

#8254 was loaded with 15.5 gr blackpowder and lead bullet.
#8255 was a Schrot = shot load
#8256 are empty cases.
A rifle power round ball load was also commonly used in smoothbores. On another page a paper shot case is shown also. This also explains the crown/G stamp on the smooth barrel: As the rifle bullet load also could be shot from the smoothbore barrel, though with shortest-range accuracy only, it underwent proof for the rifle load. It was all a matter of gun laws: Up to about 1968 rifles chambered for the 6mm Flobert and smoothbores up to 9mm bores were regarded as toys and could be used without a licence. Well, many of those "6mm Floberts" came from the Zella-Mehlis shops with a generous chamber, others wer rechambered, so the .22lr was commonly used in them. Smooth bore single-shot pistols with barrels at least 8" long also were not regarded as firearms under the old gunlaws. In 9.1x40R, loaded with the case full of blackpowder and a round ball, these were accurate and powerfull enough out to 20m and a favourite of many poachers.
I have made 9.1x40R cases from 5.6x50R brass, but the .357 Maximum would probably be better: Shorten to 40mm, expand neck with an universal expander until it fits the chamber, wind the base in front of the rim with a .2" wide strip of tape until it centers in the chamber, fire form. Use black powder only!The "service" shot load is shown on your barrel: 7.7gr black powder and 277gr birdshot.