In A Feast For Crows Chapter 24, We see Cersei having a dinner with Ser Balman Byrch and Falyse Stokeworth. Following conversation takes place:
“As I feared. Bronn is gathering swords for the dwarf. May the Seven save my little son. The Imp will kill him as he killed his brother.” She sobbed. “My friends, I put my honor in your hands... but what is a queen’s honor against a mother’s fears?”
“Say on, Your Grace,” Ser Balman assured her. “Your words shall ne’er leave this room.”
Cersei reached across the table and gave his hand a squeeze. “I... I would sleep more easily of a night if I were to hear that Ser Bronn had suffered a... a mishap... whilst hunting, perhaps.”
Ser Balman considered a moment. “A mortal mishap?”
No, I desire you to break his little toe. She had to bite her lip. My enemies are everywhere and my friends are fools. “I beg you, ser,” she whispered, “do not make me say it...”
“I understand.” Ser Balman raised a finger.
A turnip would have grasped it quicker. “You are a true knight indeed, ser. The answer to a frightened mother’s prayers.” Cersei kissed him. “Do it quickly, if you would. Bronn has only a few men about him now, but if we do not act, he will surely gather more.”
But why would she ask Ser Balman Byrch to kill Bronn?
At that time, Cersei was established as Tommen's regent and Lady of Casterly Rock. She could have formally charged Bronn of being accomplice in murder of Joffrey and send a contingent of Gold Cloaks/Red Cloaks to arrest Bronn with orders to kill him if he resists. Castle Stokeworth is right next to King's Landing so a mounted force could have traveled to Stokeworth and arrested Ser Bronn before the day's end. Then she could have executed him after a show trial.
Nobody would be expected to shed a tear for Bronn as Bronn's only aristocratic link was Lady Tanda Stokeworth, Falyse was heir to Lady Tanda and Stokeworths appeared to think that Bronn was insolent.
But Cersei instead turns to Ser Balman who mucks it all up, gets killed himself and gets poor Lady Tanda killed as well eventually.
Why would she ask Ser Balman to kill Bronn instead of using her authority as regent to dispense justice and govern in King's name? Why choose a riskier way instead of using a surer, safer way to eliminate a potential threat?
Even if Bronn asked for a trial by combat, Cersei commanded swords of puissant warriors like Ser Addam Marbrand and Ser Loras Tyrell. Ser Addam might have refused to take the part but Ser Loras could not refuse a royal command seeing as he was King's Guard and thus was sworn to obey any commands coming from a Royal provided they do not impact his first-most allegiance to the King.
Not to mention, given Cersei's attempt to get Loras killed by sending him to Dragonstone, She should be delighted to send Loras against Bronn because then she would be in a win-win situation. Cersei could even execute Bronn on spot and then let it be known that he refused a command issued in King's name thus proving his treason.
:-)She's paranoid, lost, out of her depth, afraid, rapidly losing her marbles, is convinced she's surrounded by "enemies" who are manipulating everything against her - and also, as importantly, she enjoys these games where other people do her dirty work. She thinks she's good at it.