Freeze ice to far below 0 Celsius?
Most of the cooling from ice comes from the melting anyway. That's why ice makers, which don't freeze as cold as freezers, are still useful. 

It takes 334J/g to turn ice at 0C into water at 0C, but only 2.03J/g to warm ice by 1C. So to halve the amount of ice you use to get the same cooling you'd need to freeze it to around -150C. If you're going to do that you may as well use the liquid nitrogen directly, because that's the most likely way to get ice at -150. 

If you use dry ice or liquid nitrogen in the kitchen, you need to look into the precautions (and regulations if it's commercial) that you need to take. 

You should be able to cool the sauce by transferring it to a bowl sitting in an ice+water bath (this cools faster than sitting it in a bowl of ice because of contact area). If that's not enough, here are some ideas that can be used as well 

This is done with wine and cocktails - it might work with your sauce: 


make a batch of the sauce (or a base without cream/egg etc.)
freeze it 
use the frozen sauce (base) to cool the final sauce. It may dilute ingredients you couldn't freeze but this will be much less noticeable than diluting with water. 


Sitting a bowl inside another bowl, with the gap full of ice+water, then frozen, should get you a very effective cooling device for moderate quantities. It should scale to a litre or so per vessel if there's plenty of room for ice in between. The inner container should have decent thermal conductivity and/or thin walls - a stainless serving dish would be ideal. 

Another approach is to use an ice/salt bath for freezing point depression. This can be used to make ice cream so gets really quite cold. You could precool a bowl in that and then put your sauce in it.