Not in the Budget

So, I get a call from a design house that needs something done, for which they have no expertise. Additionally, they have already set a budget for the work (with no idea how much time it will take to complete) and they come to us.

Them: "Please code out this section of the site that connects to a third party vendor"

Me: "Based on reviewing your spec, the requirements, and your timeline, this is how much it will cost."

Them: "That's not in our budget, can you do it for less"

Me: "No, this is what it will take for us to build it and fully test it. If we do it in less time we will bill you for actual - If, however scope increases the cost will also increase".

So, why am I writing this? Because I have these questions...

1. How can you possibly offer a project budget for which you have a huge portion you have no idea how to complete?

2. Why are you asking me to bail you out?

3. How would you respond if I called you and said, "I need 5 web concepts, I told the customer you would do it as a donation, that's OK right?"