Hmmm, for the feel of it, you should probably let the client do what it wants and just use the server to sanity check that what its doing is reasonable. I guess it depends on what gameplay you’re going to have in multiplayer if that is a reasonable thing to do.
Its probably going to be hard to do any reasonable interpolation between the client side and the delayed client info replicated via the server in any meaningful way. Sure, you can predict the movement by calculating the arc and perhaps try and reconcile the current position by changing the arc length (so basically calculate the expected position both for client and for update from server, then try and shorten/lengthen the arc length so that the client version comes more into keeping with the server version). But you still have the issue of firing/not firing the rope in the first place.
We got round it for our games by being turn based. So we didn’t have to reconcile the discrepancy on the client side (server essentially multicast the clients version to all other connections).