As you said, these are all the pieces already. If what I'd figure is your question it's easy to answer:
"How much damage would I do if I fire for $Length time?"
$Length * $RoF = X rounds
X rounds * $damage = gunDamage
X rounds * $DOT% = avgDotAmount
DoTs are smart in this game, they all have their separate timers and can stack independently from one another. This makes the damage as intuitive as possible:
dotDamage = avgDotAmount * ticks(element)
totalDamage = (gunDamage + dotDamage) * multiplier(element)
As this was far too easy to answer, I can only assume your question would be like
"When can I stop firing, knowing that my DoTs will kill the enemy?"
To which the answer would be "Never, unless the target isn't currently affected, then keep firing."
To us human player at least; for all the data is there in the fountain of numbers that spew from the target you are perforating with bullets. Would we be perfect computers, we could see when a new dot starts ticking(it wouldn't have a previous tick before it) and thus know exactly how many ticks are left. Being human, all we can say is "it's on fire" or "it's not on fire." This makes knowing for SURE your target is going to die impossible, the best one could do is make a normal distribution graph(wiki) with the probabilities if your target will die.
If that's really what you'd want, I could run it through some model, but it's still gonna be a pretty useless graph. Input the specific gun and skills you're using, and you could get a graph with probabilities mapped vs dotDamage. Even if you could watch that while shooting things, you can only guess how much health your target has left.
If instead, and my final guess, the question would be "How much do I overkill?" that amounts to the same, as that is also "Amount of dotDamage still to be done."
The way I have beat him in co-op was with myself as Maya and my friend as Axton. Axton used his turret with bubble shield at the start which blocks the first blasts fully and I was throwing the baby maker at hyperious with the bee equipped took him down to half health. Then you just need to keep stopping his bots from connecting to him so he can't use the blast or have his shield, we both used conference calls for this save your SMG ammo until he hasn't got a shield.
Try to get a relic to boost tediore dmg and try to keep the bots alive as long as possible but you will eventually have to kill them all. When this happens Axton should throw down turrets to attract him (hopefully, worked for us) then throw your baby maker and you should have him dead. By the time we killed all the bots he had less then a quarter of his health left so it wasn't too bad with him jumping around. Dropped some bad loot though :(
Best Answer
Various tips:
In short, use Shock or normal damage, slag him, use the Bee, release Spiderants not rats. Try to always leave a spiderant around to kill for a second wind (and so you can open the valve without washing in more enemies each time). The fight is mostly about managing the non-pete enemies to make sure you can wash off the DOT, can get a second wind but so you don't get swarmed. Pyro Pete isn't actually a big problem except for the loooong fight and the DOT effects, just keep moving and he'll never hit you with anything else. No shields spawn with immunity to both elements (unless there's a unique/legendary I'm unfamiliar with)
He can be soloed, but of course a group really does help. With a group it's much easier to manage the spiderants/rats, he's less focused on one target and it's much easier to open the valves since you can assign one person to always do that.
Here's a video demo of a Siren soloing him: