The alternative to hot air is a soldering gun style plastic welder. Mine has variable temp and a special flat spreader tip with a 1/8" passage through it. The plastic welding rod is fed through the passage right into the melting plastic under the tip. There is also a bigger spreader with no feed passage. It is made by Polyvance, formerly Urethane supply. The current model 7 is 200 watts.
I have not yet tried hot air welding but I have read that that it is best for plastics over 1/16" thick. I have not done a bumper but I have repaired a splash guard under the engine and it was pretty thick. You must either match the original plastic to get a bond or they have a great product called Fiberfix. It does not chemically fuse but acts like a thermal glue; think hot melt glue on steroids.
To make it work even better you can cut a piece of stainless screen and embed it into the Fiberfix patch. The Fiberfix comes in flat sticks that do not feed through the regular tip. You use the spreader tip instead.
A soldering gun like the Weller (with a spreader tip) might work too. There are more types of plastic than are in the Harbor Freight kit. There are tests where you heat a scrap of plastic and from how it burns or doesn't burn and the color of the smoke and the odor and the texture of the melted plastic you can identify what you are trying to repair. Also, many plastics are marked with the recycle symbol with a number inside the triangle.The number identifies the plastic. Sometimes the type of plastic is also listed under the triangle. Number decoding info is on line.