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<p>It’s not a matter of whether they had significant travel expenses - of course they do, they are coming from the opposite coast. </p>
<p>You seem to have missed the point that the O.P. would like to pick up the tab but doesn’t want the offer to be awkward. So, it’s not a matter of what you would do, it’s just a matter of helping the O.P. with what she wants to do i.e., a generous gesture without making a social faux pas. </p>
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<p>All you would have to say (if you were in that position) is ‘may we split the bill?’ If your offer is declined your all-important concerns of indebtedness are covered. And if I may be so blunt, this event is not about you, it’s about the kids who I can guarantee would be uncomfortable about a scene over a dinner tab. If you still can’t live with yourself for someone offering you something without a ‘payback’ perhaps you could do something gracious for others without expecting a quid pro quo.</p>