Mini-van taxi from Haneda - hotel in Shinjuku

There's no recent info on this subject so I'll ask my question and hope that's ok.
We are a family of 6 arriving at Haneda around 11pm, we want to pre-order a taxi/minivan that will take us to our hotel (and back when we travel home). I've tried Tokyo-taxi but they don't have any available minivans for the dates that we need.
Does anyone have the name and web site for another taxi company that might be able to help with this? I guess the other option would be to book 2 smaller cars, but I would prefer us to be in the same vehicle.
Thanks in advance!

http://www.taxi-tokyo.or.jp/english/teigaku/
The URL last part "teigaku/定額" means "fixed rate" in Japanese.
http://www.taxi-tokyo.or.jp/english/teigaku/check.html
http://www.taxi-tokyo.or.jp/english/call/kind.html
http://www.taxi-tokyo.or.jp/english/call/contact_list.html

And if not enough back to Google Search again by "羽田 定額タクシー".
羽田 is Haneda(as airport), タクシー is taxi.
Most fixed rate service companies web provided EN page,too.
Some companies using "fixed fares","flat fare" as word
and included regular "taxi service" page. (no big banner and menu on home/cover page)

Pl have a look at link. You can contact them & enquire accordingly. I've used the Kyoto service & it was good. Cabs clean, polite drivers.

MK Taxi Tokyo

http://www.tokyomk.com/en/planning.html

Thank you so much! I've managed to make a reservation! :)