Although I own all three sizes of Engel bait coolers, (and I love them), I can also understand the budget minded individual as well.
In the winter, I used to use a 5 gallon bucket with a styrafoam insert. This worked okay until you got water between the bucket and the insulator. Then it would freeze!
My boat came with mesh bags for my livewell. Although I do not use them in my livewell, I would consider using them in a bucket. I think one of the main reasons why the engel coolers are great is because of the mesh insert that goes in the box. It helps to protect the minnows from getting slammed in to the walls.
Attached is a picture of what I am referrring to regarding the bait bags. The issue however is that they are expensive (almost half the price of an Engel).
If it was me, I would go to the store and purchase a laundry bag (thin mesh). I would cut it down to about 1/2 the depth of a 5 gallon pail. I would take the top of it and wrap it over a pool noodle that is cut to go around the inside of the pail.
Fill the bucket about half full and then put in your sack. The noodle would be floating on the water and there would be a gap between the water edge and the top of the noodle so the minnows could not get out and they would be safely protected inside of the mesh bag. You could put your frozen bottles under the bag if need be.
I would drill a hole near the top of the bucket for my aerator tube. My stone would go on one end along with a large fender washer to keep in on the bottom.
Just my idea anyway.
I have OCD "Obsessive Crappie Disorder"