Best method for alligator weed?

How to Control Alligatorweed - Several Options, you might consider the herbicides rated “excellent” in Army Corp of Engineer trials.

You can email Dr. Chris Mudge, LSU AgCenter, and get his recommendation - aquatic weed control is his area of expertise. LINK

Look at the excellent/good options Crawdude posted the link to. It is a perennial so now thru mid to late October is the best time to make applications for long term kill. Carbohydrates are moving from supporting above ground growth to underground storage organs for winter survival and growth in spring. Systemic herbicides like glyphosate move readily with this carbohydrate “flow” into these storage organs resulting in better long term control. Will take several falls to accomplish this.