I understand getting duty free at Jersey and Guernsey is now difficult for private flights. You are best to go to Alderney, where fuel is c.50p a litre and they have a duty free counter which stocks the common duty free items.
Places to eat on Alderney: The Albert (pub) does good quality pub meals and has quite a good atmosphere.
Further down, near Braye Harbour, are some more restaurants, the names of which escape me and which are highly spoken of for their seafood.
A walk from the airport right down to the harbour will take maybe 30-35 minutes (downhill!) and will work up an appetite. A walk around the cliffs back to the airport will take about 90 mins and offers some good views on a nice day (France, Guernsey, Sark, even Jersey).
Route: Up to you. Either from BHD towards SKERY and into the zone from there or round via Bournemouth and direct Alderney (not above FL34 as you approach 50N). Clearance into Jersey Zone is usually not above 3000ft SVFR and you need to be aware of the visibility restrictions to operate. If you are landing at Alderney you can use THEIR met vis if that helps.
Don't forget to claim the fuel drawback on the fuel you have in your tanks when you depart Bristol for Alderney (27.34p per litre).