It seems 1-wire devices include a unique ID with them, if that's the case if theres 8 1-wire temp sensors per frame, then if you can pull that unique ID then a unique ID could be generated for the whole board by using those 8 ID's of the chips? Is this feasible?
Entirely. I'd go with the ID of the 1st device since it's unique and likely 8 bytes or more.
Have you PM'd Afremont?
Here's a USB business card that uses a surface mount ATtiny85, just so you have an idea of size.
The Tiny85 is 8 pins of what the 328P is 28 pins. The surface mount versions are all thin.
The DS18B20 is a bit bigger than a match head. They can be made right into a thin cable.
You could take through-hole AVR's, bend the legs out to the sides and connect the ones you need with female jumpers or female crimp connectors on wires. Solder if you dare, use heat-shrink if you don't. You don't need a board. You only need power, ground, a bypass cap, and whatever pins connect to your sensors. You don't need 16 MHz external crystal either, you can run on the internal 8 MHz oscillator and divide that down to use less power. Epoxy that, it won't be 1/2" thick.