I managed to track down one from Lee's parts out of Texas. Finally. These things are like unicorns. Anyway, how involved is changing one of these out? Any advice?

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Not too difficult, but you usually need a new firing pin bushing to go with the firing pin.

Remove the old bushing by carefully grinding the end until you can drive the old bushing out.
Sometimes you can use an appropriate size drill to carefully drill one end of the bushing to remove enough of the flare.
However done, the bushing will tend to rotate making it difficult to get the flare off enough so the bushing can be driven out without having to wale away at it and damage the hammer.
One trick is to put another drill bit in a vise and drilling or grinding the other end.  The bit in the vise will dig in and catch the bushing and prevent it from turning.

Clean the hammer and all parts thoroughly then apply lube to all parts

Install the new pin in the hammer and the new bushing.
Make sure the firing pin is right way up.

Expand or flare the ends of the firing pin bushing.
You can make a special punch to "roll" the ends of the bushing, or you can use two steel ball bearings.
Put a bearing of the appropriate size on each side of the bushing and squeeze in a vise until both ends are flared equally.
The flare of the bushing has to be below the level of the hammer sides to prevent any catching on the frame.
Once flared the firing pin MUST be free to move up and down easily.

They sent out a replacement nose and rivet, I'm assuming the piece you're calling a bushing is what they called a rivet?

Yes.
Just different terminology.