Considering that they apparently didn't have a strong enough first queen that they are now superseding, it would stand unlikely that they could produce any queen that would be strong enough to take over and do the job right at this point. So if i were in your shoes with the extra queen available, I would indeed go through that hive with a fine tooth comb and make sure there wasn't another queen at all, NOR any queen cups at all. Give them the mated queen and let them release her. Doing that will get your hive back on it's feet faster then waiting for them to finish making a queen, etc.