Are bottles a magnet for thieves? I suspect if you want a magic bottle, you're going to pay for it. The cheap solution is to get used to drinking warm water. Not as refreshing, but just as hydrating. When I'm thirsty enough, the water temperature becomes less important, but still, a cold drink on a hot day can be extra refreshing if you can manage it.

I have a stainless steel, double-wall Camelback bottle. I don't know if you can find it in your price range, but if you can, it might fit the bill. After the first day of riding with it, my wife asked how it performed. I said I really didn't know, because I had kept it as a backup bottle and hadn't actually used it after filling it with ice and water before setting out. I then pulled the bottle out of the bottle cage and heard the ice clink inside. This was hours later, and maybe even the next morning. I was amazed. I never worry about it getting stolen, but then if I was concerned, I'd just take the bottle with me, which I often do anyway.

Things to keep in mind:
There's a plastic version. Cheaper, but not as good.
It does great at insulating, but it's not going to cool water you put in unless there's still ice in it.
Because it's double walled, it doesn't hold as much. Less so if you fill it with ice to cool down your next refill. Probably my biggest problem is that sometimes I put too much ice in it and not enough water. The ice doesn't melt fast enough, and I end up draining the bottle and only having ice left.
The second biggest problem is the number of times I've tried to track down the errant ping that my bike is making only to eventually figure out that it's the sound of ice cubes hitting the steel wall of the bottle. On a blazing hot day, my mind just doesn't go to "ice cubes" as a potential source of the noises I hear.