However as departure time nears -- one week -- and much remains to be done, I'm looking for a shortcut here and there.
If you are pressed for time...I would absolutely do these 3:
Chapel Royal Audio Tour
Henry VIII's State Apartments Audio Tour
Henry VIII's Kitchens Audio Tour
To me, the "big story" of Hampton Court is that of it being built by Cardinal Wolsey and being taken over by King Henry VIII, who made it one of his favorite residences.
The rest, I have minimal to no interest in, myself.
Don't forget to do a walk around of the gardens and exterior of the buildings.
Grab lunch or a coffee at The Tiltyard Cafe to refresh and recharge.
To prepare for your trip, you might watch on TV some episodes of Wolf Hall, the PBS series, the episodes which deal with King Henry VIII and Cardinal Wolsey. If you have PBS Passport, you can watch any time.
It will give you a quick, concise education as to what went on as far as the transfer of Hampton Court from Wolsey's hands into Henry's.
Wolf Hall is an excellent series to watch in the evenings before you leave for your trip.
There are not many episodes, but they will give you the story of that time period.