Rather than putting the road into a tunnel why not re-route the road further to the south and hide it behind a ha-ha.
The road to could 'floated' on a continuous raft to avoid intrusion into the archaeology whilst the intervening land between the monument could be sloped at a shallow angle to the top of the ha-ha.
If the topography of this shallow slope mirrored the present topography the ha-ha itself would blend into the natural landscape and provide view from the monument similar as we are it today.
The advantage of constructing a ha-ha is that there would be no intrusion into the archaeology as all the material needed for its construction would be deposited above it.
The ha-ha would also dampen any sound from the engines of passing vehicles and if constructed at a reasonable distance away might even blanket it out altogether.
This method would be considerably cheaper than a tunnel and would also give a more aesthetic view from the monument.
Sent from my Samsung Galaxy smartphone.