Deliver the work before the delivery deadline no matter what the buyer says or does.
It is your responsibility to get the work delivered on time. Sometimes you need to ignore the buyer if they are not being reasonable, as in this case.
You are always the one in control, not the buyer. It is YOUR business on the line that matters. Do not let buyers tell you what to do. Do what you need to do to get the job done on time.
Always remember this: YOU ARE THE BOSS. It is YOUR BUSINESS. You are the business owner, you are the boss, you decide what happens.
As long as you do what your gig description says, and what you told the buyer you would do before the order was placed, then you have done your job.
Always deliver the order on time no matter what the buyer says.