The Arduino is only powering the sensor.
Why do you use an Arduino as power supply? That's a bad idea already.
For small amounts of power you can get away with it but you're dropping 7V over that tiny regulator - which can dissipate maybe 500 mW - so about 70 mA would be the limit.
Here is the circuit diagram.
Why don't you connect the sensor to your 12V supply? The Arduino doesn't seem to do anything but sitting in the way.
Eventually, I'll need to power both the Arduino and the sensor using the power supply. Do you know of a power supply part/device that I can still plug into the Arduino and pull power from it for the sensor?
Can you recommend a power supply part that lets me hook the sensor wires to it directly? I only have wires coming from the sensor, and no adaptors.