Here's a novel way to save some more money...
Seishun Juhachi Kippu: 5 x 1 calendar day per person unlimited travel for Y11,500. That's Y2,300 per person.
I'd known about these for a while - we did a 17 hour trip on it once! Limited to local trains (no shinkansen) so not great for travelling between Osaka and Tokyo - think 12 transfers with luggage - yuck!
BUT, bring on the Moonlight Nagara. It travels from Tokyo to Ogaki, which is just short of Osaka, but close to Biwako. And, not being a Shinkansen you can use the Seishun Juhachi Kippu!
Due to it being an overnight, non-sleeper equipped train, it's no good for Rachel and the boys. So they'll take the Shinkansen regardless (just under Y14,000 one way). Anyone else would be welcome to join them.
For anyone on a tight budget, we can buy one or two Seishuns - 5 or 10 person vouchers - and take the overnight train - it takes 2 vouchers per person in the Osaka to Tokyo direction. This means Y4,600 for the trip to Tokyo. Back to Biwako we could all take the Moonlight Nagara, catching it after midnight and thus using only one set of 5 travel vouchers (Rachel will stay in Tokyo).
To put this in perspective, a few comparative prices:
Japan 7 day rail pass...............Y 28,300
Minor trips not covered by pass.....Y 2,500
Shinkansen (Osaka-Tokyo)............Y 13,470
Moonlighting as Seishun (Osa-Tok)...Y 4,600
Shinkansen (Tokyo-Biwako)...........Y 11,760
Moonlighting (Tokyo-Biwako).........Y 2,300
5 days train in Tokyo (no JR Pass)..Y 4,000
So my estimate is, with the pass we'd pay around 32,000.
No pass and using Shinkansen: around 30,000.
Shinkansen up, Seishun down: around 20,000.
Seishun up and down: around 11,000
The main problem taking the moonlight train is the loss of half a day to sleep. If we went up to Tokyo this way I'd suggest going a half day early and checking into our tokyo apartment and sleep, waiting for Rachel and whoever else to arrive. This wastes a night's accomm in Osaka (Y2,600). Back to Biwako we can use our accommodation as booked but we'll be tired on our first day in Biwako (but we'll get there 9 hours earlier than otherwise planned, so hopefully can sleep). Is the sleep deprivation worth Y21,000 ($320) to you?