Not really my style. If I had to try and ‘fix’ it for a client, I’d look at some combination of path tools and flattening.
Flattening can be used to convert the lines to shapes upon which the pathfinder tools will operate.
The process would be to gather the inner content and flatten until you’re left with path objects. Then use your clipping path to remove outside points. Then you’d need to tweak the edges however you want.
I don’t have Illustrator on this comp, but I believe that the path filter works on lines as well. If so, you could copy the clipping path and use it as a ‘cookie-cutter’ on the lines. Once they’re cut to the edges you want you can just change the end caps to rounded or pointed. Remember that rounded end caps in Illustrator go past the points which start and finish a path, so do take that into account.
Personally, I like the first approach, since you can use pathfinder filters to rough up the line ends and have full control of the finished product. [Note] Occasionally, with flattening you have to try different things to get the entire selection to flatten, or to get it to flatten without rasterising. If you don't get the result you think you should, try flattening a part, removing overprint features, make sure you're not using transparency (or sometime make sure you are), and then use your pathfinding tools to build up the final project. Remember, copying objects in place is a great way to tweak sections and still maintain the artwork.