The authority wants the body to instruct Transport for London to close off the A3/ Fuller's Way North at certain times.
At a council meeting, former leader Liz Green called on residents in the public gallery to support the move.
Meanwhile, Tolworth Road is to be reopened to through traffic between Fullers Way North and Hook Road after a Covid closure.
Under an Experimental Traffic Order, vehicle access will be regulated by "No Left Turn" restrictions during peak hours.
A similar "No left turn" will also be in place for Thornhill Road.