Rex is strong as Neo said in this chapter, stronger than most people, but he's not exactly smart and he's cocky too, so one could argue he was getting impatient with finding his father, then when they're finally going to get some info Neo says they need to leave? In retrospect it makes some sense Rex got paranoid and thought Neo was acting suspicious when we take into account he's not very bright.
With this chapter I feel now it makes sense why Rex doubting Neo happened, it was basically so we could get to this chapter so Rex gets a reality check, he finally understands he's
not the top dog around, that no matter how strong he is there's people even stronger, and that if he continues to be careless and act without thinking he will literally die like a dog.
Or at least that's what I think was the intent behind their fight, I guess if I'm right Rex will start to use his brain and listen to Neo more, maybe do some training too?