We use National Express, Barnstaple, M4, Change at Heathrow for Southampton coach station, then taxi to docks or overnight accommadation. If over 60 you get cheaper deals.
I find it incredible that you found it necessary to travel to Heathrow to get to Southampton a distance of 280 miles when via Bournemouth it is 150 miles. Blooming heck as we used to say in Combe!
To our Newportonian I would say that there used to be and probably still is a very good service direct from Newport to Southampton by train and the train station in S'ton is even closer to the docks than the Coach Station although it's only a matter of a couple of hundred yards. Good luck .... Neil