I have a large butterfly bush in my back yard that has grown up between fences (the previous homeowner put privacy fence on either side of a chain link fence, so its impossible to get to all of the roots without ripping out a lot of expensive fencing). I have cut it down and my husband has put mass quantities of weed killer on them before and they just keep coming back. I have cut them down as far as I can for now, and want suggestions for what to put on them to kill them for good. There is very little other plant life in the area, except for a large oak tree about 10 feet away - no grass or other small plants to worry about killing there.
Yeah, the problem with removing butterfly bushes is that they are quite resilient. We hack away at ours every year and it grows back just as strong the next year, which I actually like because I want to keep mine.
Without removing the roots completely, there's a high likelihood it will keep springing back. I'm not a fan of things like Round-Up but Glyphosate, which is what Round-Up is, might do the trick. It is non-selective so it'll kill most every plant it touches. If you can make multiple applications of it, you may be able to kill it off. There are some other harsh chemicals you can use but I don't know much about them as we run a synthetic chemical-free farm.
If you were able to carefully disassemble a portion of the fence (which could be a pain, I'm sure) and get at those roots, you'd have a much better shot of having success.
I think if temporary removal of the fence were not an option (as it seems to be in your situation), I'd be inclined to dig down beside and under the fence where the butterfly bush is located an try to remove it that way. A LOT of work but probably the method with the highest chance of success long-term.