While I'm aware these aren't the greatest guns ever, it would be nice if I could get mine working.

Dry-firing, it works fine.

Take it to the range and put .38 Special in it, however...no bueno.  Firing double-action, the first shot went off, then something inside was locking up the action as I tried a second shot.  Even with a lot of force on the trigger, I couldn't get it to go off.

Finally, I had to manually index the cylinder back to the first chamber, then cock the hammer for a single-action shot.

That went off, then I tried the third chamber double-action again.  No go, it's jammed again.  So I gave up on shooting.

I had a lot of trouble just getting the cylinder open so I could remove the remaining live rounds.

Ammo was Aguila 130 gr. .38 Special (note the lack of +P).  My S&W 442 will shoot it all day long, so go figure.

I opened the SIA's side plate later, but couldn't see anything obviously amiss.  That's the problem: it works with no ammo in it, but locks up when actually fired.

Is this a common problem with these guns, with a common fix?  Google searches have yielded only "these aren't worth very much," and "don't shoot .38 Special +P in them."

If it can't be fixed, I guess it will be a decorative firearm only.

Sounds like a primer backing out and tying up the gun. I've had it happen with light loads. I'd try some other ammo, maybe 158s. You might also look closely to make sure the bullets aren't pulling on recoil and tying up the revolver.

Good luck.

Could be.  I'll have to try some different ammo.

Right now I don't have any .38 Special with heavier bullets that is standard pressure, only +P.

Another very common cause of this is grit under the ejector.
Unless you hold the muzzle up when ejecting cases grit and burned powder particles tend to get trapped under the ejector.
When the gun is fired, sometimes the grit embeds into the cylinder or ejector and requires some scrubbing with a stiff brush.
Often the particles are so small you may not see them unless looking hard.

The problem is the grit prevents the ejector from seating fully in the cylinder recess and that causes sticking or even jamming.

In a bad case you may need a brass toothbrush.  Just make sure no bristles are left under the ejector.
To prevent it, hold the muzzle upright when ejecting so the grit and cases will fall free.

Sadly this is not going to be one of those easy cases.  The gun is actually really clean, because it has hardly been fired.  It has hardly been fired because it keeps jamming every time I take it to the range.  I've put maybe 5 rounds through it in as many years because it simply won't cooperate.

I remember when those came out.  They disappeared in short order.  I think we may know why.  As I recall the lockwork is vaguely Smith-like, tho I doubt anything interchanges.

It is pretty much a direct ripoff.  I didn't have them open side-by-side, but when I opened the side of the SIA to see what was the matter I might as well have been looking at a slightly smaller version of either of my S&W N-frames.

About the only major difference was the large hammer spring.  Coil-on-rod type on the SIA, where it's a big leaf spring in the N-frames.

J frame Smith uses a coil spring as well.

Get some dummy rounds and dry fire it. If it still works right, then it’s definitely your ammo. If it doesn’t, then watch the cylinder gap while you cycle it. Since it seems to work dry, my suspicion is that the ammo you’re trying to use could be sticking out of the front of the cylinder, jamming up the action. Not a real common thing to run into, but I have seen it before. You may find that you need to stick with short loads.

I would get a feeler gauge. Check the barrel to cylinder gap on each chamber.
Then try this on the rear of the cylinder to the firing pin plate/Hole.

Had this happen with a brand new Ruger Super Blackhawk and the cylinder wasn't machined properly.