Julian Lewis asked Sir Craig Oliver to appear before the Defence Committee to discuss reports a missile veered off course, amid claims it was hushed up.
The MP said the ex-journalist declined, saying he had been working on the EU referendum at the time of the test.
Ministers have not confirmed or denied anything went wrong with the test.
The government has said it will not divulge "operational details" of what happened during the test but has insisted it has "absolute confidence" in the UK's Trident system.
Former defence minister Lord West told MPs he had been led to believe there had been a "minor hiccough" with the missile's automated communications system and it had been prematurely ditched into the ocean as a precaution.
Mr Cameron was still prime minister when the test was conducted shortly before the EU referendum and Mr Lewis has been pressing for details about what he and his staff knew about it.
The Times claimed on Tuesday that the Obama administration had asked Mr Cameron not to reveal details of the test. It cited sources which it said had told the paper the missile had successfully fired but encountered problems with its US-supplied software.
Raising a point of order in Parliament after the meeting, Mr Lewis said it was "regrettable" that neither Sir Craig, who was director of communications under Mr Cameron nor Defence Secretary Sir Michael Fallon had taken up his invitation to appear before his committee.
The MP has expressed concerns that, in the past, the Ministry of Defence has publicised details of successful tests but, in this case, there had been a total news blackout which could be seen as a cover-up.
"Yesterday I told the House in good faith that Sir Craig Oliver vehemently denies he or any other member of David Cameron's media team ever knew about the aborted Trident test last June," he told MPs.
"He had said that to my parliamentary office staff in terms bordering on rudeness.
"However, when he was invited to appear before the defence committee today, he told the committee clerk that he did not wish to attend as he said he had left No 10 to work for the Remain campaign before the test firing took place."
A spokesman for the former prime minister has said suggestions that his media advisers were involved in a cover-up over the test or had suggested one were "entirely false".
CNN quoted an unnamed US defence official on Monday as saying the unarmed II D5 missile - fired from HMS Vengeance off the coast of Florida - did deviate from its intended trajectory as part of an automatic self-destruct sequence.
Appearing before the committee, Lord West said it sounded like the problem occurred once the missile was airborne and the Royal Navy could not be held responsible.
"The submarine was put in the right position and was in the right mode - everything was done correctly in terms of the correct firing checks.
"The missile fired properly and went up into the air... From everything that is said, it was an issue with telemetry within that missile which, if you are not 100% certain, you do not even take a risk. That is the missile itself. That is an American issue."
The Trident system was acquired by the Thatcher government in the early 1980s as a replacement for the Polaris missile system, which the UK had possessed since the 1960s.
Trident came into use in the 1990s. There are three parts to it - submarines, missiles and warheads. Although each component has years of use left, they cannot last indefinitely.
The current generation of four submarines would begin to end their working lives some time in the late 2020s.
A guide to the Trident debate
While acknowledging he did not know what had happened, he said well-informed people had "told him things" that suggested there had been inconsistent readings from the missile telemetric system - used to record and transmit measurements and data.
"I had already been led to believe there had been a minor hiccough with the missile. Clearly I don't know but that is what I am led to believe."
Lord West said he believed ministers' reassurances over the integrity of the Trident system but they should be more open if something minor went wrong.
"It was inevitably going to come out. If anyone thought this was not going to come out, then they were in cloud cuckoo land. The Russians probably had it nicely filed away."
Theresa May was told about the test when she became prime minister in July, shortly before MPs voted overwhelmingly to renew Trident.
The Ministry of Defence said submarine HMS Vengeance and its crew had "successfully tested" but Labour and the SNP have urged the government to provide more details.
A Tory MP says a former aide to David Cameron declined a request to answer questions about what the ex-PM knew about last year's Trident missile test.