Single action firing pin problem

The cross pin is riveted in place, just like the original Colt SAA.

To repair it you'll need a small round Dremel type carbide ball cutter and a new cross pin.
You can make a new cross pin from a proper size drill bit shank.
DON'T use the original cross pin, it's already too short to use.

Remove the original cross pin and the firing pin, clean everything up.

Use the carbide ball cutter to grind a small countersink or "crater" on both sides of the hammer cross pin hole. This small crater doesn't have to be very large, but needs to be larger then the diameter of the pin itself.
But, don't cut too large a countersink or the rivet won't fill it fully.
You can put the carbide ball cutter in a pin vise and use it by hand so a Dremel or flex shaft isn't absolutely necessary.

The countersunk hole is to give the rivet of the cross pin a space to flow down below the level of the hammer surface.
The reason the cross pin doesn't stay put is because people rivet them without the countersink and the rivet is on the hammer surface.
When it's ground off to finish, the rivet is ground away and the pin comes loose.

Cut a section of drill rod longer then the hammer is wide. HOW wide is a judgement call.
Lube up the firing pin and the hole in the hammer the firing pin fits into.
Install the firing pin and the new cross pin.

Put the hammer assembly on a steel bench block or the anvil of a vise and use a small ball peen hammer to slightly rivet over one side of the cross pin using a number of taps instead of a couple of hard blows, then flip it over and rivet the other side.
Continue flipping the hammer over and tapping with the ball peen until the riveting of both sides of the hammer is complete.
If you've cut a sufficient countersink on both sides the cross pin rivet will flow down into the countersunk areas and fill the countersinks.

After getting a good rivet on both sides, use a fine file or a Dremel tool with a stone to CAREFULLY file or grind the excess rivets down nearly flush with the hammer.
Be careful not to touch the actual hammer, that leaves marks that are hard to remove.

Then put some wet or dry fine sand cloth on a FLAT surface. A small sheet of thick plate glass works, but any very flat surface will do.
Put the hammer on the abrasive, press it down and push it away from you across the abrasive a few times, lifting it off the surface every pass to reposition it on the front edge of the abrasive.
Every few strokes turn the hammer around so the other end is facing away and continue pushing and rotating the hammer.
Pushing it across the abrasive and lifting it back to the front prevents making circular scratches in the hammer as it will if you just slide it back and forth.

The idea is to sand down the cross pin rivet until it's perfectly flush with the hammer and the pin seems to disappear.
Rotating the hammer prevents sanding one end down more then the other.
This also puts the proper bright grained finish on the hammer sides.

Once the pin is perfectly flush and the side of the hammer is smooth and bright, flip the hammer over and do the other side.

Clean everything up and apply your lubricant of choice and it's done.
If you got a good countersink and a good rivet on the ends of the cross pin the pin will be very difficult to see and it won't come loose.

Yup - peen ‘da pin...

The S&W M28 hammer nose is replaced using almost the same method. But uses a lot softer rivet & more flare. More swaging then peen.

Light peening on this one.