Whether defendant Tyler Robinson acted out of political animus could be key for prosecutors seeking the death penalty.
The prosecution and defence have filed duelling documents focused on whether the crime included aggravating factors that would make Robinson eligible for execution if convicted.
Those include political targeting and whether the shooting endangered thousands of others who attended Kirk's September 10 event at Utah Valley University.
Robinson is charged with crimes including aggravated murder and has not yet entered a plea.
The 23-year-old turned himself in a day after the shooting.
The latest batch of court documents follows a preliminary hearing in July, when prosecutors presented what they said was "overwhelming" evidence against Robinson, while defence lawyers worked to sow doubt about analysts' findings.
Judge Tony Graf said he would decide if the case should advance to trial after he hears again from both sides on September 1.
In a document filed late on Tuesday, prosecutors argued the engraved bullet found in the suspected murder weapon was an "overt political reference".
They also noted Robinson allegedly wrote in a text message about Kirk: "I had enough of his hatred. Some hate can't be negotiated out."
Robinson's lawyers have argued the text message is not strong enough evidence to prove a political motivation.
Before his death, Kirk and the organisation he co-founded, Turning Point USA, galvanised the conservative youth vote to help President Donald Trump win a second term.
Prosecutors contend that Kirk's opposition to same-sex marriage and gender-affirming care stood in sharp contrast with Robinson, who was reportedly in a romantic relationship with his roommate, Lance Twiggs.
Twiggs allegedly told investigators he had considered a gender transition around the time of Kirk's killing.
A handwritten note that prosecutors allege Robinson left for Twiggs read in part, "I had the opportunity to take out Charlie Kirk, and I took it."
During the July hearing, investigators presented DNA analyses they said linked Robinson to both the suspected murder weapon and a tool he allegedly used to engrave the bullet cartridges.
Robinson's lawyers repeatedly questioned the reliability of DNA tests and other evidence from prosecutors.
