I do this regularly. This will work even better if you have two tubs as you will need to changeout the cooling water two or three times.
Depending on how much ice you have available, i would just use cold tap water for the initial cooling. If you use an ice water mix initially, you will use a lot of ice!!!! My usual routine is to drop the temp down to around 120 on the first bath w/ just water, then I will use ice and water for the second bath. If I have plenty of ice available, I will just drop the hot kettle into a huge ice bath and it works fast!!
Occasionally I have access to a commercial ice machine, I find it almost easier to grab 50 - 60 pounds of ice for chilling rather than fooling w/ my immersion chiller. This is especially true in the summer when the tap water is quite warm, and will not bring the wort down to the low 60's. I figure I am going to have to do an ice bath anyways, so why bother with the chiller at all. If you are bying ice, this method is somewhat cost prohibitive. I have also culled the ice from my fridges icemaker in advance of brewday so I have 2-4 batches of ice from my fridge.
Everytime I read someone someone saying that a chiller is mandatory equipment and ice baths don't really work, I kind of chuckle. With enough ice and a big enough tub, water/ice baths are very effective! More effective IMO than any one stage chiller w/ higher tap water temps.
It also helps to have a nice large oversized kettle, You will want to have the ice bath at a level that when the pot is submerged the cooling water is a few inches below the rim of the kettle, and the kettle is stable sitting on the bottom of the tub, so there will be no cross contamination.