I’m sure you have though about it already, but if you had a way to modify the refractive index of the resin, you could make the fishing lines disappear completely.
Edit: A quick search reveals that there is quite a lot of prior art from the optical community. Darn, maybe I'm not done here.
I've never heard of index matching fluids! Will have to read up on this.
Also, if the plastic can be "invisible", you can also house it in invisible plastic- thus creating, a exploded working model, inside the resin