You're right Julie that this plant can take just about anything, but it is quite large and wouldn't want to induce heavy dieback given its prominence.
Plus, there's that "I've ripped out too much damn plastic/visqueen weed barrier" aversion
, that makes me loath to add any to the landscape.
Organic Weed Control
I've used a vinegar based product called Burnout with minimal success. Decent foliage burndown, but quick return of the culprits.
Scythe is considered "mostly organic", as it is based on naturally occuring Pelargonic acid, and provides excellent burn-down. Repeated applications would be necessary to control perennial weeds, and the purists won't be appeased...
I've found this site useful for sorting through the options.
There's no organic magic bullet, IMHO. Organic weed control is only successful when deploying a multi-faceted strategy, that would include: 1) minimizing soil disturbance, 2) controling/eliminating seed sources, 3) fostering desirable species, and as a last resort 4) selective herbicide use. And in some situations, Glyphosphate might be the only, and best, recourse.
What I object to is the very anti-IPM practice of "Spray and Pray", which comes about because a client insists that "can't we just spray something on it?" I have seen that approach fail too many times to believe it is anything but a panacea.
For instance, take a Chem Lawn maintained yard off regular broad-leaf herbicide applications, and admire the quantity and rootstock size of the dandelions that return by mid-Spring...
After all, if the Chem Lawn and Scotts programs actually killed the weeds "roots and all", they wouldn't be a successful franchise or retail product, would they?