I went for the low cost solution and took the burner out and plugged about a third of the burner holes with brass screws.
I had a similar problem on my Brinkman All-in-one. The flame would blow out unexpectedly (from a breeze) if the burner was turned down low but I needed to turn it down low to stay at the desired temperature. Plugging up some of the holes was a great solution.
To do my mod, I used some automotive muffler repair paste to block some of the holes in the burner. The paste comes in a tube and hardens like cement after heat is applied for the first time. I separated the two parts of the burner (which are held together with a bolt) and then filled about 1/2 of the gas channels that the gas goes through with the muffler repair paste and then bolted the burner back together. Now only about 1/2 of the holes have flame but these flames are twice as big and don't blow out. One nice thing about the "muffler paste" plug method is that you can not see that any mod was done to the burner when it is turned off. Of course, run the burner for a while without any food in the smoker until the paste sets. This would avoid any toxic gases that may occur (if there are any) when the paste is first hardened.