Throwaway will be best I think. Open jaw cheapest around $1.2k

it can be something like JFK-LHR-JTR then book LON-JFK

play around for what is cheapest - e.g. Oslo-JFK in October , you don't care if you aren't using it !

if you just want a cheap flight then Norwegian/ Easyjet have a flight on 3rd June on easyjet.com - it's part of their "alliance" so the transfer would be protected, though you need to collect & re-drop your bags at Gatwick at a special desk