I found two books which have chapters or sections on Mt. Rainier National Park, and while, the NP isn't the focus of the book, there is some interesing information about the NP, besides being good reads too. One should be easily available at most chain and some small bookstores and one probably will have to be ordered if you want a copy.
The first is "The Snow Tourist" by Charlie English, a reporter for the Guardian newspaper in the UK. He travels the world to talk to people living and working in the coldest places where there is snow, lots of snow. From near the artic circle to South America.
He has one chapter on the most snowfall in a given year, either Paradise at Mt. Rainier NP if you want to adhere to data collection standards, or Artist Point (ski resort) at Mt. Baker some distance north of Mt. Rainier and just south of the Canadian border, which has the "official" records but it's debated as they didn't use accepted standard method to determine the snowfall and snowpack.
Personally, I'll take Mt. Rainier's side on this issue because it's done at a site which is federal government reviewed and approved. It's not on the map of sites for Mt. Rainier NP because it's not a real-time site, meaning it's daily readings by NPS staff at a specific location. Both are described in the book.
The rest of the book is also excellent about the other places in the world where snow dominates life and work.
The second is "Wilderness in National Parks" by John C. Miles of my alma mater, Western Washington Univeristy. It's the story of wilderness designation in national parks. It's a historical and geographic tour of the NP's, where Mt. Rainier was one of many to have wilderness designation in the NP, meaning no commercial developmernt, such as roads, facilities, etc.
The book only talks about Mt. Rainier for about a dozen pages, not a lot in the 300+ page book, but the information is interesting and he confirms an idea I read in another book but not elsewhere. The 1915 map (field work 1910-13) was not just about producing the first map of the NP, but for the engineering purposes of building a road around the mountain, about where the Wonderland trail currently is around the mountain.
The road effort was never finished due to the obvious engineering and financial hurdles it would take to complete. And shortly thereafter a large section of the north side of the mountain, between the Ipsut campground in the northwest corner at the end of the Carbon River trail (formerly a road) to the current sunrise visitors center in the northeast corner from the White River entrance.
That area was the first wilderness designation for a NP and sealed the fate of any road circumnavigating the NP. Which adds to more of the firsts with Mt. Rainier NP (but the 5th NP). It was also the first designated from existing Forest Reserve, the Washington Forest Reserve. The road didn't happen, or at least all of it, but the map persists with the 1938 and 1971 updates.
Anyway, if you don't want to buy the book, get them in a bookstore with a cafe, buy a cup of coffee (I'll take a doppio con panna if you please) and read the sections or chapters.