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You don't want the southside, it is either mud and dirt on the river path or a boring ride over the B349. Plus you miss the cool dark section after Barnes Bridge.
Just cross Barnes as usual and ride to Hammersmith Bridge.
Ride the river path towards Putney until it spits you out on either Woodlawn or Stevenage. At the end, turn left at the T-juction then right into Fulham Palace Road.
Over Putney, back immediately on the Fulham Railway Bridge and continue as before.
River path surface on both sides is fine for riding as long as no recent rain (Barnes side can get a bit splashy), but will be slow with peds/kids depending on the time you ride it...