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I finally had it with this old leader-line library I was using so I gave it a couple of hours tonight and wrote my own method. It was honestly way easier than I thought it'd be; I just needed to learn how SVG works (a bit embarrassing I haven't really done this until now) and everything fell into place. It's made nice and efficient by Svelte's reactivity, so only the lines that attach to a box that's moving rerender. Feels very smooth.
Next up is figuring out a cleaner way to do dynamic links, as my old implementation was a hack because of the weird way the library I was using worked.
One thing I might also think about is adding some kind of bezier curve so the lines look a little nicer.