The .44 British Bulldog was a centrefire round. Its easy to check on your revolver if its RF or not.
However the name bulldog became generic for pistols of that type and there were .44RF “Bulldogs” made. The most common (in the US) being Hammond. Almost certainly a few Belgian made copies too under various names.(or no name)
If its a Hammond its probably more valuable than the British varients, and older. Depending on condition of course. Look in the barrel and chambers, any sign of pitting? many of these pistols have never been fired because they were pocket pistols carried daily but not fired. Trouble is if they were fired they were usually never cleaned properly because the sort of people who bought them didn’t know how. If there is a ring of pitting in each of the chambers in a circle where the bullet goes thats not from firing. Its where the tallow on the OL bullets has attacked the steel.
The round is so feeble it wont open a can of beans. If it is a RF don’t fire the primers off in the pistol because they are corrosive, and your pistol will require a full clean, thats if they go off at all.