In that case, with the fact you said it would take about 15 seconds for him to die from withering, even if Rex does blow him up, they would both die.
Let's say Rex rushing in mechanical sword blazing. Wither gets him, and he resorts to long range. If wither goes down, wouldn't Rex still die?
Allow me to tackle this in a few ways.
1. Wither's powers work in the form of continuous contact. On a normal person, even temporary contact would be pretty bad, but as noted, Rex has enhanced regenerative abilities. That means that if he can break from Wither's grip in time, while he still has things that can be healed, he'll be able to survive.
2. What do you mean by "blow up" exactly?
3. If he rushes in with his BFS (though his usual preference is his giant fists, but we'll ignore that), then yes, he'll be able to damage Wither, but his sword will also begin to break down and potentially move to his body. There'll likely be some form of pause from the realization of Wither's nature, which will give Wither a chance to rush him. It isn't like he can jump super far or anything, and it'd take time for him to create his mechanical legs or jetpack to get away. From a vampire.
4. The actual rushing with the sword. His sword isn't really suited for thrusts or stabs, due to size. It's slicing exclusive, or crushing can be argued. That means that it isn't particularly fast anyways, making it a bad matchup against a normal-looking human sized opponent.
Edit: Ninja'd.