Have you tried 'painting' it? Put some Round-up on a small paint brush and apply it to the ivy. Time consuming but better than some of the Round getting on things you want to keep.

Ground ivy says "oh please don't put down mulch" like brer rabbit said "please don't throw me in the briar patch." I've been fighting ground ivy for 22 years. Don't leave it on soil or mulch after you pull it. It will just reroot. You can put it on something it can't grow on and leave it in the sun. When it is totally dried out then you can compost it. It doesn't seem to produce many seedlings. It doesn't need to. If you leave a tiny bit of stem or root behind when you pull it it will be back. I haven't used Round-up on it. I just pull it and then it comes back bushier. If you don't have half an acre of it and you think you can get rid of it by painting it with Round-up I'd say go for it. Show it no mercy. I must have spent thousands of hours pulling it with no end in sight.

If you spray on a quiet morning with a shield (paper bag, poster board) you can work pretty close to existing plants. I've been known to hit a dandylion in the grass on my way to the garden- just bend over and with the nozzle on stream get it on the weed.

Ground ivy is a big nuisance, I agree. But in my opinion, using Round-up to get rid of ground ivy would be an even bigger nuisance. We just go around and pull it regularly. Then pull it some more. Then pull it some more... Maybe just try to make it a habit to pull it whenever you see it?