Under normal circumstances, Rui would most certainly have gone for the stamina victory. Not only did he possess better stamina due to having a more balanced spec configuration as far as his Martial body went, thereby placing a greater weight on stamina, but he also possessed the Final Breathing technique that would allow him to preserve much more energy in the long run.
Furthermore, Rui's Forestep technique, which he was now forced to use at a maximum was less energy-consumptive than Kane's Godspeed technique due to it being imperfect.
That meant that the battle would normally turn in his favor as time passed and Kane tired until he exploited an inevitable opening to end the fight.
It wasn't glamorous, or flashy. But it was a surgically ruthless path to victory.
The issue was that they were no longer fighting under normal circumstances. They were fighting under special circumstances.