Howdy
I can't guarantee I can help you, but I too suspect you do not have a #3. The Number Threes were big guns, just as big as a Colt Single Action Army and were usually chambered for 44 caliber cartridges. 45 in the case of the Schofield.
If you post a couple of photos it might help.
Yes, the sideplates on the old antique Smiths are removed the same way as on a modern S&W revolver. Do not attempt to pry it off. Remove the grips. Be careful if you have the old hard rubber grips like this because they get brittle with age and break easily. Don't over tighten the screw when putting the grips back on, you might crack a grip or pull the threaded collet out of the grip on the other side. I suggest whacking the grip frame with the wooden handle of a hammer. Hold the revolver with the side plate up, and keep your thumb on the side plate so it does not pop out and fall on the floor. Strike down on the grip frame with the handle of the hammer. Inertia will cause the side plate to work up until it pops free.
I'm wondering if maybe you have a 38 Double Action. 38 S&W caliber, five shot cylinder.
It can be tough to identify that model without something to compare it to for size. The 44 Double Action (top), 38 Double Action (middle) and the 32 Double Action (bottom) all looked very similar, but varied greatly in size. The 44 was a six shooter, pictured here with six 44 Russian rounds, the most common chambering for that model.The 38 is pictured with five 38 S&W rounds and the 32 is pictured with five 32 S&W rounds. The latter two are five shooters, not six shooters. (P.S. The 44 Double Action was the only double action model built on the big #3 frame. There were several variants, but there were no other double action Top Breaks built on the #3 sized frame. The other #3 size Top Breaks were the American Model, The Russian Model, The Schofield Model, and the New Model Number Three).
I drew a line across the grip frame of the 38 Double Action to show where to strike the grip frame with the hammer handle. Do not use anything metal, a nice hickory hammer handle works well. The 'screw' in the middle of the side plate is actually a nut with a slot on it. It screws onto the hammer pivot stud and keeps the side plate in place.
Here is what that model looks like on the inside with the hammer all the way down. The thing that looks like a claw in front of the hammer is the double action sear. It rotates the hammer back when the trigger is pulled double action and once the hammer rotates back far enough the hammer slips off it and falls. The single action sear is a separate part. You just barely see it poking up towards the hammer near the bottom of the hammer. The 'half cock' notch on the hammer is visible near the single action sear, the full cock notch is up near the pivot for the hammer stirrup (the part the hammer spring attaches to).
The hammer has been drawn back to 'half cock' in this photo and the single action sear has popped into the 'half cock' notch on the hammer.
Full cock and the hammer is all the way back. It is being held in place by the single action sear, barely visible engaging the full cock notch on the hammer. To tell you the truth I have never taken one of these apart further than this. It is very common for the full cock hammer notch to be worn on these guns. Or the single action sear can be worn or broken. Either condition will prevent the hammer from staying back when brought to the full cock position.
Good luck finding parts.