As others have posted, there is a N'Ex every 30 minutes. If you miss the 3:15 pm, just take the 3:45 or 4:15 to Tokyo Station. That said, I think you and your family, along with all the paraphenalia associated with traveling with young children, are a perfect fit for the Limo Bus. It will take you to the Conrad with no need to fight your way through Tokyo Station with luggage, trying to find a cab. The cost is pretty similar and the traffic hassles aren't that bad. It is not like you would be driving. Just relax and enjoy the trip.

I'd suggest staying in Kyoto rather than Osaka. JMHO.

I just don't see the "ryokan experience" as something for young, noisy children. Again, JMHO.

I agree with the above recommendation to avoid flying domestic HND-ITM. It can't be that much more expensive for a oneway trip from Tokyo or Shinagawa to Kyoto after you subtract bus fares from ITM and all the hassles of flying domestic and the shinkansen is just far more "cool" for kids.