Can you describe a real-life situation when drawing a case-study proposal (as opposed to a standard one) is the only way to win a potential customer?

As I understand from your question that some times to draw from the submission or don't to submit will attrack your customer.

Yes it's great tacktics & idea but you need to play it in correct way as the following:

1- you should trust that you are biggest player in the area/business of your customer

2- You should aware about the inside of your customer's environment

3- Inform your customer informal way that you can't submit for this RFP. You should know that your customer needs your company services/name/expereinces to implement this project

4- You have to be ready if the customer ask you to submit then you should ask him for your requirement