Suzanne,  if herbicides are not an option then the best strategy is to increase the competitiveness of the forage stand.  There are herbicide options other than glyphosate (ie 2,4-D or dicamba).  These other options will not kill the grass forages and would not require reseeding.

Is this an existing pasture?  Is it a productive pasture, does it need management to increase its productivity?  Does it need fertility or lime? Is the stand adequate or will it need some reseeding?  Managing creeping charlie (common mallow or malva neglecta) by improving the stand will take some time and areas of the stand that are thin or less productive will give a chance for the weed to continue to grow.

feel free to email me directly if you have more questions <personal data hidden>

Suzanne, I am glad Mark VanGessel reached out. Our other ag agent also posted these links you might find useful: 

“First make sure it [creeping charlie] is correctly identified as what they say it is. If any chemicals are used, which they say they can not use, make sure the label is followed. Here some extension websites:” 
https://apps.extension.umn.edu/garden/diagnose/weed/broadleaf/creeping/creepingcharlie.htmlhttps://hort.extension.wisc.edu/articles/creeping-charlie/