Explanation:

- "Offered": This implies voluntarily proposing to do something, often based on one's own initiative. It doesn't quite fit as the sentence indicates determination rather than a proposal.

- "Suggested": This means to propose an idea or plan, typically less forceful and more advisory. It's not strong enough to convey the determination to pay.

- "?? Insisted": This is the correct word. It denotes a firm determination or persistence in doing something, which aligns perfectly with the context of paying the bill despite potential objections.

- "Requested": This involves asking formally, which does not convey the insistence implied by the context.