Rampage/Gunlust Gunzerker. Shock Infinity Non-Elemental Infinity (because Pete is Fire/Caustic so I assume he has resistance to those) Alkaline Bee (Having either corrosion or fire resistance is nice, but either way you're going to get hit with a DOT) Legendary Gunzerker mod Sheriff's Badge Quasar grenades (Useful for corralling the enemies that come out of the pipes.)
I just sprayed away until he does the unavoidable (at least as far as I can tell) AOE that will both burn and corrode you. Every time he did that I activated one of the valves on the edge of the area and ran under the stream of water to clear the DOT effect. From there I cleaned up the enemies that come out of the pipes and go back to blasting away at Pete.
That's about all there is to it really, his attacks are all very easy to avoid aside from the unavoidable DOT. I would recommend prioritizing the valve that releases spiderants first, and keep using it until those mobs are level 60. Then switch over to the valve that releases the human "rat" enemies since they can drop your Bee shield with some random gunfire, whereas the spiderants are pretty easy to avoid.
I'm sure there are much more effective builds, but this was just the build I was playing around with while running through the story missions so I decided to try it out on the raid boss.
Green Lantern of the Comics Board I have never watched Dr. Who.