I don't know much about welding plastic but here's my two cents for what it is worth.
Years ago, we used to boil the nylon propellers from our model airplanes. This was because the plastic would get brittle with age and boiling the plastic softened it up again. Our boiled propellers could handle much more abuse than the non-boiled old props.
I suppose you could compare welding plastic to welding aluminum in a sense. Most materials will weld much better when you get them closer to the temperature that you will be welding them at.
"So, my advice, short and sweet is to try to preheat your material prior to welding it." In the case of a plastic bumper, try to heat it up with a hair dryer prior to welding it. Once you've made your weld, it will probably look pretty rough.
While the plastic is still sort of soft, try planing it smooth with a rigid ribbed body file.
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've use these body files for shaping the plastic mouthpiece on my saxophone. Then, finally sand it down with fine grit sandpaper (200-600). You can use toothpaste as a final finishing polish. We use it on CDs when they get scratches in them. Don't knock it, it works...