I have inquired with one of professor Kato's collaborators, professor Moriyama from Osaka University, who could give some information on this 1978 manuscript "*On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields*":

> 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 <A HREF="https://www.jstage.jst.go.jp/article/pjab1945/52/4/52_4_180/_article/-char/ja/">On an explicit formula for class-1 “Whittaker functions” on
 GL$_n$ over $p$-adic fields</A>, but there are much more complicated
> case-by-case arguments.

I have contacted the Tokyo library, hoping that they can make a scan available.