We found that often early afternoon (11AM, noon, 1PM ET) worked out well. Remember that every day during the afternoon (early morning Japan), the site is down for what I recall is 3 hours for maintenance.
Over the past year, since we planned a trip there, we were on the web site dozens of times. We almost always got a page holding us in a virtual queue before we got to the site, but never had a problem loading. Not to question what you're doing, but make sure you have the correct site (as I expect you do)..... Perhaps download the TDR app and see if you can get to their reservation system through that. Otherwise, I'd keep trying. They were often busiest early in the month when TDR vacation packages went on sale, but I think this being the end of the month, that shouldn't be an issue.
Good luck - hope you can get in and do your planning. It's worth the effort!