The army has refused to halt its offensive until the group completely surrenders, but the fighters have vowed to fight to the death.
In addition to the soldiers killed, an undetermined number of fighters and more than 20 civilians have lost their lives.
Meanwhile, the al-Qaeda-inspired group confirmed in a statement posted on Wednesday on an Islamic website that its deputy leader was killed some days ago.
The statement said Fatah al-Islam's deputy commander, known as Abu Hureira, had been killed, but gave no details as to the circumstances of his death.
It said the group was bringing the "good news to the Muslims of the martyrdom of a noble brother".
"We, in the Fatah al-Islam movement and all the true mujahidin, pledge to God to avenge our brother's blood from the infidels and the renegades."
The authenticity of the statement, posted on a website which commonly carries messages from al-Qaeda and other fighter groups, could not be independently confirmed.
On Monday, the government announced that police had killed Abu Hureira, a Lebanese also known as Shehab al-Qaddour.
Ghazi Aridi, the country's information minister, said he had been killed a few days ago by police in Tripoli, near the camp which is located on the city's outskirts.