I tried something different with this, i am sure I could have made it work if I started it earlier or if I had got the idea sooner.
I used my light tent with a black light over the top. No other light.
I opened shutter with black light on for about 15 seconds turned off black light reached in and took out the thermometer then bounced a speedlight off the white background once from each side of the tent. I think two more shots with the speedlight would have provided a better WB.
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This could have much more potential if the background and rug were true white. The "color temperature" (white balance) is off - too cold (blue). Clever idea nonetheless. Good luck in the challenge.