How to stop pooled object's position jumping for clients?

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I've been working on an RTS game using PUN Classic.

I'm having some trouble with pooling projectiles. When they are reused by the pool they correctly re-activate and position themselves correctly on their owners client, but for the other clients the projectile appears for a single frame in the position where they deactivated.

Their movement is handled by their owner, with Position + Rotation Transforms synced and also Rigidbody with Velocity and Angular Velocity. Interpolate is set to Lerp and teleport is on.

My feeling is that the other clients are trying to interpolate between the missiles last position before deactivating and it's new position once respawned. Does this seem likely?

(Is this just a terrible way of handling multiple projectiles anyway?)