Hi again,
Hmmm - honestly you're going to have to pack a lot in in the few hours you're in Hakone to get it all done. If it was me I would stay overnight - find a good hotel that has an onsen and relax for a night. That way you can make sure you get everything done that you want without feeling frazzled! For me personally I don't like to feel like I have a timetable to keep to - but I know you are on a timeframe so..... ultimately it's up to you. Which is more important? Getting to Osaka or spending a bit of chill time in the countryside? That might be the easiest way to make that decision
As to luggage on the shinkansen - be warned that there is very little space and you're not allowed to put your luggage in the aisles. The best advice I can give with Japan is to travel with as little luggage as possible. It's seriously not fun dragging heavy bags round with you when you're lugging them up and down stairs/train platforms etc - especially when you have little ones with you.