To help with help clear up Kylo’s affiliation and the “killing” question, we could potentially go with something similar to this line change I suggested over on the Rey Nobody thread. (Ignore everything before it since it is Rey Nobody-related)
Kylo: What Palpatine doesn’t know is that we’re a Dyad in the Force, Rey. Two that are one. Together, nothing will stand in our way.
Kylo: Together, we’ll take the throne.
Though the first option might be better if you don’t want to associate Kylo with the Sith throne. Plus, these lines also might work well for the eventual Rey Nobody version of Ascendant.