Hi Bernie, yes, that is me and I do live in Shikoku, but down here in Kochi, so I don't have any knowledge about cycling in Kagawa, or any of the shops. I've only done one event that way, and that was just a crit that the Kagawa Cycling Federation holds every October/November around Lake Basho (?).
Hence, all I can recommend is to visit your local main cycling shops and ask there. Here, 2 out of the 3 main shops organise rides every Sunday, but they can be a mixed bag, as it depends on who turns up. But they are all open to anyone who wants to join, and I never had a problem even in the days when my Japanese was worse than it is now. If you want faster rides, then if you can prove yourself to the other riders, they might invite you to join other more targetted rides. Also, there are some teams here, which are organised separate to the shops, so you could enquire about those, but they might be invitation only.
The other option is to have a look on Strava, and that might give some indications as to where you can find rides. Here, I just use it as a way to find routes for riding (I've never uploaded anything).
Also, there might be some local ALTs who cycle, so maybe you could ask through the local Kagawa AJET page.
If you are interested in doing events, then I can give you a list of the main events that take place in Shikoku, but there aren't that many. For instance, I am not aware of one actual open road race you can join.