Pump beer thru immersion chiller placed in ice water?

Trying to balance water usage/cost...
It took a while (and a lot of water) to cool my last batch.
Can I make an ice water bath in my HLT and place my immersion chiller in there...then pump the wort thru...back into my brew pot until I reach my desired temp?
It would be a balance of ice usage vs water usage. Don't want to spend $20 on ice.

Maybe get a second immersion chiller, than set the first one in a bucket of ice water, that will chill the tap water, than the second one goes into you wort.
Pumping your wort through the chiller seems kind of risky to me, but if you are confident that your chiller is clean and sanitary if and you can do it without oxidizing too much that should work.

I have been pumping through my herms coil to chill. Ground water is about 53 f now and I throw some ice in. I have a bunch of old silicone bowls I fill with water the night before and let freeze.

My old copper chiller could chill to 68 in just over 10 minutes using about 12 gallons. With the Hermes coil I'm using about 8 gallons is ice + water but it takes 15+ minutes.

6 gallons of wort BTW and my point was I think it's worth trying! Could always run test with just water.