Jack did a great job describing how to do it using propulsive engines. I have a different answer:
We already (almost) planned to do it (inadvertently).
The original plan for the recently launched Parker Solar Probe was to do a gravity assist at Jupiter for a subsequent fly-by of the Sun at a relative speed of more than 300 km/s. So, to get to a speed of 370 km/s instead would require a distance to the Sun of 3.5 instead of 4 solar radii - totally doable if we don't have to deal with sensitive instruments that need to be shielded from the intense radiation and heat.
Now we just have to make sure that the velocity vector of the probe is properly aligned with the CMB, but this is possible: The inclination can be varied by aiming at different edges of the Sun while the direction within the ecliptic just needs a proper timing with respect to the position of Jupiter.
Unfortunately, this maneuver provides the "being at rest w.r.t. the CMB" requirement for the "small experiment" only for one instant in time, and not for an extended period. If you need that, we're back at Jack's answer.