You will likely find the best accuracy is derived from a laser-based measurement. My Bosch is laser only, no ultrasound component and has 1/16" resolution. For your application, the six inch minimum distance limitation of the Bosch is outside your specifications.
Because laser measurement devices use time versus distance, it's possible you could construct a mirror chamber with front-surface mirrors that would cause the laser to travel six inches before exiting the chamber, overcoming that limitation. The calibration should be straight-forward, as you'd place the chamber flush and adjust your mirror placement until you read the exact expected compensation distance.
You would need only small mirror panes, which would not "break the bank," but if you find a discarded large-screen projection television, the mirror within is front surface and quite large. Front surface mirrors should not have contamination from touching and should be cleaned with care, as you can erode or otherwise degrade the reflective surface.
I found this site:
which looks promising for a different model of laser rangefinder, although the creator has run into a stumbling block or two.
It would appear that the key is to overcome the minimum distance limitation, perhaps as I suggested above, and interfacing with serial communications of the arduino.
One of the links I pursued involved the creator contacting the technical support of the manufacturer. The support ticket was closed as "resolved," in the manner of advising the user that the warranty was void by opening the case. Yeah, like that matters to makers and hackers!