The eight-year-old is currently 60th in the weights for the Aintree race on 9 April with the top 40 guaranteed a run.
He travelled strongly under Richard Johnson although Katie Walsh challenged briefly on Broadway Buffalo.
But Johnson and the 13-2 shot jumped well and slogged their way up the run-in to win by nine lengths.
"It's a dream to have a runner in the Grand National and that's where he's going," said Lee.
"I don't think the heavy ground suited him, but he'll go to Aintree whatever we get.
"It's been amazing and started when Grey Gold won on Hennessy day and it's been relentless. We've got a lot of nice horses."
BBC horse racing correspondent Cornelius Lysaght:
The run of rookie trainer Lee, who's now won significant races on five of the last eight Saturdays, was described by her father Richard, from whom she took over last summer, simply as "a-mazing". She is certainly making her mark.
There were concerns beforehand about Bishops Road's stamina over this marathon distance, especially in the mud, but he dispelled them all with a nine-length win, after which jockey Richard Johnson suggested the horse would be better still in drier conditions.
It's Aintree and the Grand National next.
Bishops Road showed stamina in tough conditions to give in-form trainer Kerry Lee victory in the Betfred Grand National Trial at Haydock.