I don't understand why you'd want to cut out Turnham Green on the District when its got its own platforms. Most economically the Ealing Broadway and Uxbridge branches only of the Picc could serve Chiswick Park on the outer tracks with a new junction just west of the District divergence at Turnham Green. The Heathrow trains would stay on the middle pair and avoid Chiswick Park, as shown here:
Indeed, it seems as though @LU_timetabler began typing with Chiswick & Turnam the right way round, but then swapped them by mistake? Like so (potential corrections in red):
The Piccadilly will eventually take over Ealing Broadway service from Ealing Common. The District will then not serve Chiswick Park or Turnham Green. Chiswick Park will become a Piccadilly served station. Plans unclear for Turnham Green - possibly it might be relocated onto the Richmond branch.