Professor Atsushi Murase from Kyoto Sangyo University was kind enough to send me the preprint. I am uncertain whether I can post it publicly, but if you write me an email I will forward it to you.
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Professor Tomonori Moriyama from Osaka University gave me some background information:
The paper you mentioned is the master thesis of S-I. Kato under the supervision of Prof. N. Iwahori, which is hand-written and written in Japanese. I read it in the library of Graduate School of Mathematical Sciences of University of Tokyo more than 20 years ago.
The basic strategy of proof is more or less the same as T. Shintani's 1976 paper On an explicit formula for class-1 “Whittaker functions” on GL$_n$ over $p$-adic fields, but there are much more complicated case-by-case arguments.