Negotiate an extension on the deadline and explain the reasoning, producing work provides a bad representation of the company.
No wonder you did not get the job. You need to talk to your manager and ask your team for some help. There are deadlines in every industry and negotiating to extend the deadline with the manager is pretty much useless as the client will not be ready to wait... :/
You are the owner of a clothes shop and someone enters with a £50 voucher. You notice that the voucher is one day out of date, but the customer has exactly £50 worth of clothes that they wish to spend, what do you do? Accept the voucher or not?
I said that I would accept it as it is only 1 day out of date and this way the customer goes home happy and will probably return.
I would advice the customer about the date and depending how the customer reacted, I would ask my manager if I could offer the customer £25 instead of £50. That way the business makes business and the customer goes away happy.