Actually, it was a real battered woman's shelter (terrifying experience that was actually pretty calm, I just don't like being in fenced in areas I can't leave of my own free will). Onto your points:
-Despite this being a fight where their general fighting morality isn't totally present, the conditions of the fight aren't to kill, they're to win. That means that to an extent, we can still use their personality traits, which is why I list the ones that are relevant, and not full on personality analysis.
-Rex has shown that he'll rush on unknown enemies all the time. This is a stupid behavior trait, but do note that all he knows about is EVOs. Just the same as the previous thread, they have no knowledge of each other's abilities, and with that in mind, Rex could assume Wither is just another EVO. Rushing in would be his tactic until he established what Wither's abilities are, in which case his intelligence would kick in and he'd switch to ranged combat.
-Rex is established to defeating enemies without killing intent, but with intent to remove their ability to fight then attempt to cure them. Whether he'd attempt to cure Wither or not can be argued, but the fact remains that Wither would be out for blood for the get-go, so the beginning exchanges would determine the battle. If Wither gets a good grab on Rex, then he can effectively shut down enough of Rex's body that he can't use his nanites to regenerate it. If he loses a limb, he loses access to some of his weapons, and if he loses some essential chunk of body, he'll just die altogether.