Ohashi-junction & Yamate-tunnel by public bus

Hello!
As already asked in this community one year ago,
I am also highly interested in traveling through the
very interesting Yamate-tunnel (山手トンネル)
and the Ohashi-junction (大橋JCT) in Tokyo.

As a private Taxi is too expensive, we are planing to use
that "Airport Limousine Bus" from Haneda to the city.
So my question is:

Does the bus route from Haneda to Shibuya run through
the new Yamate-tunnel (C2) beginning at Oi and then
leaving the tunnel using the famous Ohashi-junction loop
turning eastwards on the Expressway No. 3 to Shibuya?

Or:
Which bus route do you recommend, to see a part of
the Yamate-tunnel together with the Ohashi-juction loop?

We will arrive at Tokyo next week, so If anybody can
help me, that would be very nice.

Thank you / arigatou gozaimasu

Currently, Haneda - Shibuya limousine buses don't run Chuo-Kanjo Shinagawa line(via Ohashi - Oi JCT) eventhough the line is opened in March 2015, this is because, the highway entrance in Shibuya is not installed nor opened. The entrance will be installed and open in 2018, at this moment, the exit is installed already but not opened yet. I suppose, both entrance and exit open in 2018, Shibuya.

Currently, Haneda - Shinjuku and Haneda - Ikebukuro limousine buses run Chuo-Kanjo line(via Ohashi - Oi JCT) as schedule. BUT, limousine buses must run on time due to flight schedule. I suppose 99% of buses on the route run via Ohashi -Oi JCT, not a 100 % guranteed.

Hello & thank you very much for your answer!

So, we will take the route from Haneda to Shinjuku
instead. But does this route run through the
Ohashi-JCT loop tunnel (going in circles from
underground to bridge) or does it only pass the
junction, continuing straight on C2 direction north
to Shinjuku?
If we take the bus route at the later evening
(aprox 9-10 pm), traffic should be not so bad?

Best regards

From my past experience, limousine buses between Shinjuku/Ikebukuro and Haneda don't run Chuo Kanjo line(C2), via Ohashi-Oi JCT and Yamate tunnel.
Since Chuo Kanjo line(C2) is fully opened in 2015, traffic jam was much reduced for both private and public transports. Therefore, limousine buses don't need to run new route despite they have permission to go through. Once C2 is fully packed, or involved in road accidens, there are only limited exits C2, thus limousine buses are unable to avoid traffic jam. They must run on time always, this is the reason that limousine buses don't choose to run C2, I guess.

All limousine buses are controlled by the traffic center by GPS, traffic and drivers reports. They always choose the very best route to avoid traffic jam and road accidents, limousine buses have almost all road permission include highway, tunnel and public roads to avoid traffic jam while the other public transports(ie, local buses) don't have road permission to go through.

The othe route I can think of is Kichijoji and Kokubunji routes.
Haneda - Shinjuku/Ikebukuro limousine buses use Hatsudai south entrance and exit to get in/out highway, which doesn't go straight into C2, whereas Kichijoji/Kokubunjii routes use Shinjuku JCT which go straight into C2. BUT, Kichijoji/Kokubuji routes are limited and the route is chosen by the traffic center so is gamble to choose the route. They won't use C2 on Kichijoji/Kokubunji routes, I'm still unsure.

Thank you very much again for your help, I will just try it.
Best regards