Hieracium aurantiacum is a rhizomatous perennial in addition to being a prolific seed producer so it's pretty unlikely you will be able to control it without herbicides. It's one of those plants that is best hammered when it first shows up. Delaying aggressive control just results in an exponentially more difficult problem later.
I see that Milestone (aminopyralid) is one of the recommended herbicides for Hieracium aurantiacum control. Milestone is an interesting herbicide. It's a broadleaf killer like Tordon (picloram) or 2-4-D but, unlike most of them, is somewhat selective in its effect and kills plants from some families but not others. Milestone has been touted as a "miracle herbicide" for the control of invasive weeds in natural areas. It's NOT, but it does seem to cause less collateral damage than most. Whatever you do, please don't use Tordon near the stream. Tordon is a persistent chemical and mobile in the soil column and will have all sorts of disastrous consequences, both for the native plant community and for aquatic life in the stream.
