The Addicks accepted an offer by a group of investors represented by Varney in early December.
The Addicks' third defeat in a row leaves them two points off the play-off spots.
Rommedahl was the Addicks main threat and gave Fulham left-back Franck Queudrue a torrid first 45 minutes.
The Addicks lost 2-0 to relegation rivals Nottingham Forest on Saturday and are eight points from safety.
The Addicks are owned by chairman Richard Murray's Baton 2010 Limited, who took control in the summer.
Michael Doyle almost drew the visitors level when his overhead kick was blocked by Addicks defender Patrick McCarthy.
Addicks : What we see converging in the new world is the emergence of mass and direct marketing.
He twice denied Addicks striker Nicky Forster and also superbly saved shots from Deon Burton and Nicky Bailey.
He was a West Ham academy trainee before leaving to join Charlton, and went on to make 169 appearances for the Addicks.
Borrowdale, 24, joins on emergency terms after the Addicks suffered injuries to Kelly Youga, Grant Basey and loanee Johnnie Jackson.
BBC: Charlton Athletic snap up QPR's Gary Borrowdale on loan
The 23-year-old, who has made two substitute appearances for the Addicks this season, will stay at The Valley until 2010.
The Addicks needed to win to have any chance of survival going into their final game of the season at Liverpool.
The Addicks are searching for a new manager following Alan Pardew's departure at the weekend after they lost 5-2 to Sheffield United.
He played nine games on loan for Sheffield Wednesday in 2006 before joining Charlton last August and appearing 38 times for the Addicks.
BBC: SPORT | Football | My Club | Liverpool | Liverpool include Carson in plans
The Addicks deservedly led when Hayes found Scott Wagstaff, who rounded keeper Stuart Nelson and produced a fine finish from a tight angle.
Bishop scored in bizarre fashion just after half-time when Alan Gow's free-kick deflected off his backside and into Addicks keeper Rob Elliot's net.
It was a 10th defeat in 18 games for the Addicks this season and Pardew was sacked after less than two years in charge.
Heatherson became even more frustrated when she headed wide either side of the break, but Aluko finally put the Addicks ahead in the 58th minute.
BBC: SPORT | Football | Women's Euro 2005 | Charlton Women 1-0 Everton Ladies
Brighton took the lead after Charlton failed to deal with a corner, Gary Dicker crossing for Inigo Calderon to slot past Addicks keeper Rob Elliot.
Newcastle midfielder Amady Faye is also in talks with the Addicks.
The Addicks had lost six of their seven Premiership matches heading into this London derby, but were by far the better side in the first period.
It is hoped that the money will help the Addicks, who are currently second in League One, win promotion to the Championship at the first attempt.
Dimitar Berbatov scored a superb solo goal inside the opening 10 minutes and former Charlton player Jermain Defoe sealed the Addicks' relegation with a powerful shot.
The 24-year-old signed for the Addicks in 2007 and has been on loan at Crewe and Gillingham and had been strongly linked with a move to the Gills.
Addicks striker Joe Anyinsah then went close at the other end with an attempted lob as he was knocked over by Nelson, who was booked for his rash challenge.
Charlton Athletic Football Company, which owns the players, and Charlton Athletic Holdings, which owns the Addicks' stadium The Valley and the training ground, were transferred to Murray's new company in July.
And, although Bishop's name is on the scoresheet, it could have just as easily been chalked down as an own goal after the ball deflected into the net off Addicks defender Simon Francis.
On 3 December the Addicks confirmed that lifelong fan Murray had accepted a legally binding offer for the acquisition of Baton 2010, Charlton Athletic Football Company Limited and Charlton Athletic Holdings Limited from an investor group represented by Varney.
He scored five goals in 37 games for the Magpies but the failure to hold down a regular starting place led to a move to Charlton where he scored three goals as the Addicks finished 13th in the Premier League in his first season.
应用推荐