I have been taking our 1 year old Cockapoo in the Mog's passenger seat to get him used to the wind and noise and he just curls up and sleeps.
Unfortunately he has grown to twice the expected size and is all legs so is not a welcome lap passenger, Yet!!!

However, He has had one 10 mile outing with us to West Wycombe Classic Car show (photo below).
He will be going to Lay Hill Bucks with us on Sunday Afternoon (A nice Country Village Green Classic Car Show) as its not too far.
My next plan is to see if he will fit into and take to the Back Shelf (with internal extension).
We have a Seat Belt Harness but could do with a Single to Double adaptor of some sort. Plus a Rear Shelf Belt Clip if that one works.
The biggest issue is that most of the places we like to go (National Trust - Brooklands etc) won't allow Dogs.