If that's the main conflict, I don't think this is that complicated. But let's go over some questions.
1. Frisk and Kira can both just LOAD/RESET or Bites the Dust if either of them dies, right? Likewise, can either of them do this while Unconscious? Your body being gone is one thing, but if it's still intact but not functional enough to think?
2. What are the statistical values for their AP, Striking Strength & Durability of our combatants? I know the OP said Frisk has higher determination than usual (& more Determination means more Durability.), & killing might not be feasible... knowing how well they match up can give us an idea if they can knock one another out. Presumably, they're pretty close, with Killer Queen being 8-C & Frisk being 9-A.
3. They both remember their experiences of scenarios they SAVED/LOADED/RESETED or Bites the Dust-'ed, right?
Supposing the answers to 1 & 3 are yes, there might be a conclusive option.
If Kira remembers his experiences, he can trial & error. And unlike Frisk, he has a stand, with its own set of limbs.
Frisk's Lifting Strength is: Likely
Regular Human (Able to carry the Annoying Dog)
Kira's Lifting Strength is:
Athletic Human,
Unknow with Killer Queen
As Yoshikage Kira, Kira was able to lift his hand while it was weighed down by the ability of Echoes Act 3, which... well, our Wiki says
"the perceived weight of it was 40-50 kilograms when reflected back onto Yoshikage Kira." IIRC, Kosaku Kawajiri is a pretty big man, so he might of similar strength to Kira's original body. With that weight, he walked to Koichi.
Frisk probably lifted the Annoying Dog with both hands, which the Undertale Wiki speculates is a Samoyed breed. Google tells me males of that breed can weigh 20 to 30 kg.
So, if Kira in this form can maintain a lifting strength of 40 to 50 kg on one hand with difficulty, & Frisk needs both hands to handle an average of 25 kg, Kira might be able to hold Frisk's arms in place with his own hands.
By holding Frisk's hands in place, he could prevent Frisk from punching, kicking, using a knife to stab, etc. And the size of their limbs is probably great enough that he could do it with enough distance to avoid being kicked or headbutted by the kid.
Sure he might not have as great an AP/SS as Frisk, but he has the Lifting Strength to keep them restrained, & he might need only moments for Killer Queen to act. It gets slightly easier if they can bring the kid near unconsciousness, too.
From there, Kira just has to use his extra set of limbs: Use Killer Queen to deal a knockout blow, such as by striking the back of Frisk's head, or cover/pinch Frisk's nostrils & cover the kid's mouth to deprive their brain of oxygen so they fall unconscious. 'course, they'd have to be careful not to kill Frisk, or they have to start all over again.
With enough trial & error, Kira might be able to work out a plan like this. From there, it's just a matter of fighting skillfully enough to actually achieve it, which could be inevitable if they keep killing each other & starting over.