Tex absolutely didn't want to betray Renault, so when he found that there was a feeling between Renault and Kerrigan, he tried to convince Renault all the way, "This woman murdered 8 billion lives and doesn't deserve a second chance. Kill her!" He has no guilt about killing Kerrigan per se. And for him, it also means freedom. The only thing that stands between him and freedom is his friendship with Renault.
Note that Tex really has to kill Kerrigan in the end, because his life is in the hands of the emperor. If people talk to Matt after robbing Odin, they will find that there is a remote-controlled bomb in his clothes, and the emperor can take his life at any time if he wants. In the last scene, Tex had to shoot. If Tex really had no respect, he wouldn't have hesitated so long and told Renault about it.
in the end, Tex did die. When he returned to bn interface after customs clearance, it clearly said so. Of course, the possibility of returning in the future is not without, for example, the body is controlled by Zerg infection, and there are many ways. And Tex's bullet went through Renault's shoulder. If you pay attention to the last scene of CG, you can find that Renault has a big hole in his left shoulder and is still smoking.
Of course, by the way: I think even if Tex really killed Kerrigan, according to the emperor's personality, he wouldn't let him live.