He wouldn't get cancer before the fight starts, he'd only have a chance of getting it once the fight starts, passives only start when the battle starts, so Metal Bat only has the possibility of getting it post-the battle starting, to which his Battle spirit would keep him alive.
And he'd work through cancer as it is only made when your cells are damaged and begin to multiply without end, this takes a long while to become fatal, usually months, Additionally, Cancer isn't the most common effect of radiation poisoning, it's just the scariest because we cannot cure it, let's compare to the Elephant's foot:
"After just 30 seconds of exposure, dizziness and fatigue will find you a week later. Two minutes of exposure and
your cells will soon begin to hemorrhage; four minutes: vomiting, diarrhea, and fever. 300 seconds and you have two days to live. "
None of these effects would phase Metal Bat, he's lost gallons of blood before and continued, which is worse than hemorrhaging and fatigue. And it takes 500 seconds, or about 4 minutes to give you 2 days to live based on that, but that's not gonna work on Metal Bat while he's fighting.
And Godzilla hasn't the AP to put Metal Bat down in one go either, as Metal Bat has taken 18 Megatons of damage and kept fighting (His fight with Garou again)
And while this happens, Godzilla is getting the shit kicked out of him while Metal bat gets more and more pumped to kill this freak.