Here is what I would do. I would drill carefully the back side of the boulder sign rivet on both old and new. Just enough to take the curled head off.
I would use the old sign on the new Boulder.
Then use a similar diameter size screw (#6) and on in the inside use a #6 flat washer and Elastic stop nut. Riveting plastic takes special tools and novice will break expensive plastics every time.

I have had the recent pleasure of playing this awesome game at the Crofton Bowling Alley in Maryland. Even though the playfield and ramps are grey with dirt, the furry guy does not move, one of the pop bumpers is Nono, it still plays well, even with a credit dot!

So with interest, I called the company that it belongs to and asked to buy it. The owner stated that it was a very collectible rare game that he would sell for $9995. I responded that if it was that valuable, then he should have someone take better care of it and told him of its issues. He laughed and said have a good day.

So the life of a routed machine is sometimes doomed.

I can relate as I am trying to build a Doctor Who Pinball from parts. Lucky for me that I have a fully functioning one to reference. I have been sourcing WPC89 parts for a year now. The Transformer was an expensive hard to find, but I ran an ad. I got two people sending me pictures. I bought the first one, as the second one was asking why I needed one and wanted way more money.

I printed out the complete parts list off of the ipdb website ( https://www.ipdb.org/files/2768/Williams_1993_White_Water_Parts_List.txt ). I used a highlighter to color similar named parts on that list. Like MS - stood for machine screw, for example.

The lamp boards were hard to find, but after 6 months, I found a set in France and one in Argentina. Helps to have part number to google and find it on Ebay or Pinside in search. The hardest part for me now is the correct ball guides. Unless you find someone who recently salvaged the exact game, the unique parts are a quest for a holy grail. It looks like all the ball guides were custom made for each game, like the plastic ramps and wire guides.

Good luck on your search and be persistent.