Their 31-14 home loss was their 10th in a row and now their only hope of survival is if the Championship title winners - to be confirmed following play-offs late next month - do not meet Premiership entry criteria.
Newcastle are clear favourites and as a top-flight club last term it appears inconceivable there will be a reprieve for Welsh.
It has been a fraught season for the Exiles who were docked five Premiership points for fielding an ineligible player this season.
Major shareholder Kelvin Bryon has publicly declared his intention to cease his funding at the end of June, while intense speculation had surrounded Copsey's future as chief executive.
Now he has confirmed his departure from the club, something clearly painful for Copsey and a decision that appears to have been dictated by economics.
In a statement on the club's official website, Copsey said: "I'm obviously very disappointed to be stepping down as the chief executive of London Welsh, but given the situation I did not feel carrying on in my current capacity was the best for me or the club.
"I will continue to support the club going forward and wish it all the best in the very challenging times it faces ahead.
"I know it has the players, the staff and the spirit to overcome the difficulties it has had to contend with in recent weeks and would like to thank all the staff, players and supporters, who have been tremendous in their support and professionalism through increasingly difficult times."