Narita-Haneda with more than 10 hours

We have a flight transfer in Tokyo, from Narita to Haneda with about 10 hours to kill in between. We are wondering if we would be able to take a quick tour to Shibuya (a picture with Hachiko statue and a bowl of udon somewhere would be nice)?
Our plane is supposed to arrive on a Wednesday at 2:00 PM in Narita (2nd week of August). We have a connecting flight leaving from Haneda at 1:00 AM (Thursday). So that’s about 13 hours in between.
We are a family of 6 with 4 big luggages and 4 small ones (carry-on). One person is on a wheelchair! We figure, the most efficient way is to take the airport limo from Narita to Haneda. Store our luggages at the airport and catch a taxi or train to Shibuya.
Can somebody please offer us any input on this plan? We are on a budget, so would prefer cheaper options if it’s available. Our concerns are:
1. luggage storage at Haneda
2. transportation to and fro Haneda-Shibuya (cost, ease for wheelchair, time, rush hour problem)

Thank you!!!

It should be the proper way to move all the luggage from Narita to haneda first and take a train back to the city. How much is the shuttle between the two airports (i am curious)?

Another budget way is to catch the Kansei line from Narita to Ueno (1000yen per person) and store the luggage at ueno. Ueno itself is something to browse around and have something to eat. Then you can take train to shibuya or shinjuku for a walk (by yamanote line). After dinner, go back to Ueno for the luggage and take the yamanote line again to shinagawa station(400yen to 500yen from Shinagawa Station to the airport) for the train back to Haneda. The travelling between Ueno and the Haneda airport might be a bit hassle, but since you should be travelling after busy hour (after dinner at 9pm or so), you should be fine.

"Another budget way is to catch the Kansei line from Narita to Ueno (1000yen per person) and store the luggage at ueno."

First of all, the name of the line is Keisei, not Kansei. The commuter train (1,000yen) option is not recommended for tourists with luggage and especially for someone in a wheel chair.

If it were me, I'd take the NEX to Shinagawa and store luggage there. Shinagawa to Shibuya is 12 minutes via the Yamanote line. Buy the Suica/Nex combo at the airport. The NEX fare will be 1,500 yen. You can use the Suica for Keikyu Line. You can also use the Suica as cash at many stores.

Shinagawa to Haneda is 20 minutes by the Keikyu Line.

Yes, 1."it should be the proper way to move all the luggage from Narita to haneda first" by airport limousine-most time effective with a wheelchair passenger and 4 huge +4 small bags.

2. checkin luggage, collect boarding passes at the counter. stow carry-on at lockers. With check-in luggage on board, the aircraft will not leave without the pax.

3. Catch a train- Tokyo monorail to Hamammatcho- with lots to see/dine. Catch the same train return to T1 or T2 at Haneda.

4. Collect carry-on bags and wait for flight.

This route will be considerably less stressful with 6 pax, no worries on missing connecting trains or luggage, no language issues and arrive at the right terminal.

Thank you all for your input and suggestions! Lester, the airport limo Narita-Haneda ticket is 3000 yen.
Donald, would we run into rush hour problem if we head out to Hamammatcho station from Haneda around 4:00 or 5:00 PM?

"would we run into rush hour problem if we head out to Hamammatcho station from Haneda around 4:00 or 5:00 PM?"

It will be closer to 6.30PM after clearing customs at Narita, limo trip, checkin Haneda etc. Rush hour will not affect your family on the forward journey (except allows you to view closeup rush hour crowds when entering central) as your single line trip originates from Haneda; you will have a seat. On your return journey late evening to Haneda, crowds should have dispersed.

Note: Airport limo accepts yen or credit cards.

Hope it helps. Have a memorable few hours in Tokyo.

Thank you Donald for your help! We are really excited. Wish we have the option to stay a few days, but are grateful for this little time we have there. I really appreciate your input!

Reminder: Reserve wheelchair so the airlines staff will have it ready at Narita airport from plane to the limousine level, and at Haneda airport from limousine to terminal to train. He/she should be able to use it on his/her trip downtown for a few hours. If queried, show the officer your boarding/transit passes.

Enjoy!