Can this contract continue and OY 1 exercised if both parties agree under a bi-lateral? Can a J&A under FAR 6.302-2 Unusual and compelling urgency or FAR 6.302-1 Only one responsible source and no other supplies or services will satisfy agency requirements be used to make this happen?
1) Yes, bilateral exercise of the option is a typical course of action for the scenario you described *** AND *** it is necessary to obtain an approved Justification and Approval (J&A) for Other than Full and Open Competition (criteria in FAR Subpart 6.3) because modifications to the contract in question are now out-of-scope (option was not exercised in timely fashion per clause FAR 52.217-8 and the performance period lapsed). Your HCA may allow the practice of a J&A Amendment instead of issuance of a separate, new J&A.
2) Obtain legal review from your procurement counsel to ensure your HCA doesn't have specific policies prohibiting this approach.
3) It's a best-practice for the requiring activity and contracting activity to maintain a "reminder" of approaching contract option periods to avoid lapsed contracts, so please consider following this best-practice.