Obama ordered prosecution proceedings to be frozen while his administration evaluates the war court.
The US president has already said that the prison camp will be closed by next January, but legislators remain unsure on what to do with the jail's 240 remaining inmates.
US military prosecutors say they have viable cases against 66 detainees at the prison.
A delay until September in prosecution proceedings is needed "to address and eliminate all possible challenges to this [prosecution] process", Captain Seamus Quinn, who is prosecuting al-Qosi, told the military commission.
But defence lawyers say that the government had played politics with the lives of Kamin and al-Qosi, who have been detained at Guantanamo for years.
"You cannot sit somebody in indefinite detention. It violates every principle we have as Americans," Lieutenant Commander Travis Owens said.
Kamin is accused of joining al-Qaeda, training to make detonators for roadside bombs and transporting weapons to be used against US forces in Afghanistan.
Al-Qosi is accused of being a bodyguard and driver for Osama bin Laden, the al-Qaeda leader.
Khadr is accused of killing a US soldier with a grenade in Afghanistan seven years ago.
Canada agency criticised
Khadr's defence team told the hearing on Wednesday that he did not wish to be represented by military defence lawyers.
The Canadian wants to fire his entire defence team made up of American members of the military, and wants to be represented by Canadian lawyers, Al Jazeera's Monica Villamizar reported from Guantanamo.
As the military commission convened, a Canadian review panel reported that the country's spy agency ignored human rights principles when questioning Khadr at Guantanamo.
The Canadian Security Intelligence Service (CSIS) also failed to give full consideration to possible mistreatment of Khadr by US officers at the prison, the report said.
"Security Intelligence Review Committee believes that CSIS failed to take into account that, while in US custody, Khadr had been denied certain basic rights which would have been afforded to him as a youth," the report, said.