JRiver MC definitely has a Disc # field and it uses it. I'm not sure how it's interacting with how your files are already tagged. I *am* certain that JRiver obeys the Disc # field and sorts properly.
You should take a look at the full tags of a few of the songs in one of the albums that you are talking about. Highlight one song and then go to the Tagging Pane. Right click and select Tag, or just press Alt-Enter. Once the Tagging Pane opens on the left, you might need to make it show all tags. Click in the upper left corner of the Tagging Pane and select "all tags". Then find the Disc # tag and see what is in it.
If it's empty for all tracks, you'll need to fill it in. Once you have it set correctly, all of the songs will sort properly by disc # and then by track #.
If you can select all of the songs in disc 1 at once, you can tag them all in one operation. Just type a "1" in the Disc # field in the Tagging Pane and it will set them all at once. You might be able to sort by filename or by import date or something to group them together for easy selection. Or just manually select them (using Command-click) individually.
Good luck.
Brian.