I'm not a fan of changing from Piccadilly to District line at South Ken; it involves escalators, stairs, a couple of long schleps - meh

The change from Piccadilly line to District line is easiest at Hammersmith, where it's a walk from one side of the platform to the other. Can be done at Barons Court as well but the platform is narrower (and more people will have joined the train at Hammersmith so it will be more crowded).

If you do take the Underground to Victoria (much easier than HEX plus the long way round on the Circle line), sit at the rear of the District line train as it's nearest the way out at Victoria station.