I regularly use Chinese pinyin input methods with Cool VL Viewer on Mac without problems, but unfortunately my ability to help with Japanese is limited because I don't know that language. I had never thought to try to log in to Second Life using my Chinese input method (QIM). Yes I find that having QIM on blocks entering my avatar-name and password, but does SL let you choose a Japanese writing name and password in the first place? Once I have logged in to SL, I find that the OS/X built-in Japanese input method works in chat. At least I can use the Hiragana input method, and the few words of Japanese I know can be selected in Hiragana and kanji. They display correctly on my screen too. Sorry if I'm using the wrong terminology, but I don't know Japanese.
Oh... I see that the OP is using El Capitan. I'm still on Yosemite. There are known problems with Japanese support in SL under El Capitan, because Apple changed the default font files that they offer, and LL has not updated the list of fonts in SL's configuration file. SL is looking for a file called 华文细黑.ttf which is no longer there. Here is the JIRA ticket:
http://jira.phoenixviewer.com/browse/FIRE-16318. Yes, that is a Firestorm ticket but the same problem affects all viewers I believe.
I should perhaps say that I live in Australia so the "primary" country keyboard set-up I have is English. I have Chinese input method set up so that I can toggle into my native language with the standard Cmd-Spacebar.