Lobbying registrar Alison White last month confirmed that MPs and peers who accept money to influence ministers will need to register as lobbyists.
It followed the cash-for-access scandal, in which former foreign secretaries Jack Straw and Sir Malcolm Rifkind were recorded discussing the possibility of working as consultant lobbyists.
In a spirited response, Ingham said: "Given that MPs are barred already from lobbying for profit, this piece of advice falls into the 'interesting but utterly irrelevant' category of life.
"I confidently predict that not one single MP will appear on this register for as long as it exists. If they do, I will eat the register in its entirety, in Shepherd's restaurant in Westminster, the few pieces of paper it entails softened only by a glass or two of the House of Commons' cheapest house white."
The cost of registering on the scheme is expected to be about £750 per year, it was revealed last week.
The statutory lobbying register is due to go live on 27 March.