The nanny's attacker turned on Lady Lucan, beating her severely before she managed to escape and raise the alarm at a nearby pub.
Lord Lucan's car was later found abandoned and soaked in blood in Newhaven, East Sussex, and an inquest jury declared the wealthy aristocrat was the killer a year later.
What happened to Lucan remains a mystery and he was officially declared dead by the High Court in 1999.
Mr Bingham, who was in the house with his siblings at the time of the attack, said it was "extraordinarily unlikely" that his father was the killer or paid somebody else to carry out the atrocity.
He believes his father lost all sense of perspective as he became increasingly worried about being blamed for the nanny's death.
He told the Daily Mirror: "I think Dad felt backed into a terrible corner. I think he chose almost immediately to take his own life.
"He had such a huge sense of pride and couldn't bear to consider the horrendous storm that was coming. It was his intention, therefore, to vanish ... and vanish for ever.
He added: "Dad adored boats. He even built a powerboat. As a seaman, he would have known that if you jump from a boat in the English Channel, you will bloat, float and be washed up with the tides."
"It seems very likely he would have had access to a small motor boat somewhere in Newhaven harbour.
"He would have got on board with a bottle of whisky and some pills and taken it out to the 50 metre mark, the point where if you go down you're not going to come back up again, but not so far out that you are in the shipping lane."
The former merchant banker has said he would prefer that to trying to understand why his father had left the family for "no apparent reason".
Mr Bingham said at the weekend: "I've always thought it extraordinarily unlikely my father went into our family home, wandered down and killed anybody with a piece of lead piping for the love of his children, while those very children might well have come downstairs and witnessed this appalling carnage."
He also dismissed the possibility of a contract killer being involved, but added he had no idea of the extent of his father's involvement or his guilt.
Following his disappearance, there were reported sightings of Lucan in Australia, Ireland and South Africa.
His brother Hugh Bingham said he was confident Lucan escaped the UK to begin a new life on the African continent.