If it were me, I would just cut as close to the fence with the lawn mower as I could and then dig/pull out the rest of the weeds.

I’m a senior and tried that. They keep growing back because the roots are on the neighbors property

Roundup?

Spray the weeds with an appropriate herbicide. Just keep the dog away until the herbicide has dried.

Are you sure the chain link isn't a shared boundary fence? In which case the neighbor would probably be glad to be RID of it. Why else would they have put up that gorgeous wood fence except to hide the chain link (and perhaps you - don't take it personally!). A conversation with them lauding the wood fence may yield a perfect solution. Explain that you won't be able to control the weeds which will surely make their way through the new fence to their side anyway. Nor will he be able to maintain the new fence.

Both fences belong to the unfriendly neighbor. Tried being nice, unfortunately didn’t work. So I avoid and ignore them. I’ve mowed and pulled those weeds to many times. Going to try vinegar, salt and Dawn dish detergent weed killer. Saving up for a fence. I bet he will take both his down when I put up my fence and use them on his other boundaries. Can’t use round up or any harsh chemicals as I love my precious little dog. And he has a nice German Shepherd I wouldn’t want to harm. Or maybe sell and move away. Neighbors, ugh! Thank goodness for fences! Lol

The RoundUp or other commercial herbicide will NOT harm your dog!! Or the neighbors' dog!! If you fertilize your lawn, your are very likely causing your pets more problems than any herbicide would. Just allow the herbicide a couple of hours to dry on the weeds before allowing the dog free rein in the yard

Vinegar does not effectively "kill" weeds - all it does is cause desiccation of the foliage, leaving any roots still healthy and intact. To actually kill the the weeds, you need a commercial herbicide that will translocate down into the root system. Vinegar and all other homemade concoctions will not do that.