Just off the top of my head I would say that it looks a lot like Hagley Hybrid, also known as Pink Chiffon. Here is a link that might be useful: Hagley Hybrid

mnwsgal, yes I think there will be room for both to grow in your lilac bush, mine is a tree though and has long limbs that the rooguchi needed some encouragement to climb so I added some treated mesh to lightly wrap around the lower limbs and help the clem grab on and drape where I wanted it to. The nodding bell flowers look so pretty dropping down now that the lilac is finished blooming. You may want to thin some of the bushiness from your lilac so you get to see all the beauty these clems have to show. :)

With some additional effort, you can make it "climb" horizontally. You should pinch off the tips of new growth and train it along the chain link fence. When you pinch of the tip, at a leaf node, two new tips will start developing from the leaf node. Keep on doing this and you will be able to train it to fill the length of the fence and it will have a much "fuller" look. Don't wait till the buds emerge.

Yes it is. Perhaps Betty Corning. Please add a photo if they open more. Betty tends to stay as a bell like that, but there are others that start that way but open almost flat eventually.