"We need to take your recorded statement before we can move forward."
Suggested response: I am happy to provide a written summary in the next 48 hours. I will not give a recorded statement at this time, and I am not required to under my policy or yours. Please email me confirmation of the claim number and the adjuster of record.
Why this works: No state requires plaintiffs to give a recorded statement to the at-fault carrier as a condition of claim payment. Recorded statements lock in admissions before the medical picture is clear; written summaries do the same factual work without the risk.