Centipedegrass is pretty finicky. It is sensitive to different herbicides and sensitive to too much herbicide (think overlaps, etc). It is also sensitive to winter kill and doesn't go fully dormant in most of the US, so I don't like to treat it with anything while it's not fully green. I've even seen half rates of herbicides labeled specifically for centipedegrass do damage when applied when the lawn is off-color. My best advice would be to use this as an opportunity to get the customer to switch to a better turf species.
As for glyphosate, it has no root uptake. It is taken up through translaminar activity, then translocated throughout the plant. I can't stand the misnomer about "kills to the roots." Either it kills or it doesn't. Glyphosate interrupts the shikimic acid pathway, which allows the plant to make different armoatic acids, specifically tyrosine, tryptophan, and phenylalanine.