Have you ever tried to kill liriope with Roundup? I have and it's pretty darn hard to do. I'm sure the label won't tell you this but a low rate of Roundup will work, I have never tried it on mondo but I would'nt think there would be a difference. I used 1 and 1/2 oz. per gal.
Am not familiar with southern problems, but do not sell the Roundup possibility to the customer. I know of situations that I can use roundup to clean grass out of other ground covers, but always sell it as a total kill, and replace the groundcover. Then when the groundcover survives, the customer is pleased with the much reduced cost. To sell a Roundup treatment as a selective control is against label directions, therefore is illegal.
Since henbit is an annual, it would be most functional to find a preemergent you could use, and apply in a timely manner in the future, so post-emerg control is not needed.