You may be missing the leaf return spring that activates the single action pawl: I have one like that. One of these days I'm going to try to get around to fixing it. If you open the sideplate and move that pawl away from the the hammer it will perhaps function single action couple times until friction, etc. moves it back in towards the hammer. Without the return spring it won't stay in its proper place. (Note I'm not sure my terminology is correct on this part and hope you can understand what I'm saying. Plus I'm hipshooting from memory which gets even worse as I get older and older: and I have dirt beat now!)
These are fascinating revolvers just from the standpoint of mechanical variations.
I think the external hammer block safety tends to be found only on pre-WWI revolvers as I suspect they were difficult to keep clean, etc and they were vulnerable to breakage: I have a folding trigger variant with external hammer block with a broken upper arm that provides the block between the frame and hammer. But I'd have to wonder if there can be any definites on the Bodeo revolver and have to seriously sympathize with any Italian ordnance people who had to work with them. I believe these are the LEAST standardized general issue military handgun I'm aware of.
Older references like Hogg and Weeks "Pistols of the World" reference 60 variations plus. I haven't seen that many and am sure I never will but I have well over 5 in my own collection, including one with the older 1872/1874 type Chamelot Delvigne lockwork, and one with boss on the frame through which the ejector swings out from, rather than the more conventional ones that are attached around the barrel as it joint the frame. You're already well aware of the 2 major differences in the hammer block safety as well as the trigger guard vs. trigger guardless varieties, and there are the shortened barrel variants made in the 1920s. One of which has a frame mounted firing pin (love to see one of those though I have seen them for sale on net sites.)
Prices on Bodeos are all over the map. You did well if you got one for under $200 though they're out there, but I've seen them sell on internet auction sites for well over $400. These are another revolver where the pre-1899 dates seem to be a real "plus" in what they'll bring.