You seem to be all over the place in what you're trying to do.
I don't see anything about how you tried to download the subs.
Did you try both remux and Extract to SRT file? You should definitely try Extract to SRT to see if you get error(s).
If trying Extract to SRT file or if you already downloaded, you can use option Subtitle Only.
Also, if using Extract to SRT do you get an error or what happens?
NOTE, not all Amazon movies have srt, but it's a very short list. Although I use Amazon US it might be the JP sites has fewer selections that have srt.
If you can't get it working, then the only other thing is telling SF support that you're having these issues on the JP site.
Click to go to > Feedback/Support
As there is no issue in US and I have NO access, nor would anyone else, unless they also use Amazon JP site.
ALSO note the reply by Germania (here) it seems to indicate some of what you tried may have not had srt available.