If your landlord would allow some sort of mesh to hang down or if you can place some hooks between the slats to support twine or string, then that might be an alternate to a trellis. The danger is someone coming along and snapping the vines. Ivy has always been a tough one to randomly snap, but it's also an aggressive vine. Perhaps a virginia creeper might do if it can be initially supported somehow until it is established. It may infact be able to attach directly to the wood fence (which might cause its own issues).

I'd thought about using fishing line or plastic twine -- much harder to tear than regular garden twine.

The vines I'm planting are all annuals -- I don't mind th fnce being bare in the winter!

You could do fishing line. There were some threads a few years ago on the Balcony forum where the posters used eye hooks to thread the fishing line through and secured the ends with a ferrule that can be crimped closed. Twining vines when guided to this can generally climb most the rest of the way on their own.