The main trick is to zero the laser rangefinder to the ship's forward axis, then use it to aim at the processor.

To zero: Laser range a distant rock, and lock the rock in the target computer. Change pitch and yaw to zero the target indicator, and point the laser at the center of the rock. You could also do this without a rock by rolling in place, and pointing the laser at the center of the roll, where it no longer wanders among the stars.

To aim at the processor: with the laser now zeroed, aim a bit above the center of the entrance of processor, because the rock you throw travels several meters below the axis of your ship. Also, add about whatever Z-height you are about to drop to get to tether range on the rock.

Lower your ship, tether up, throttle up, give the rock some speed and cut it loose*.

* obviously you can cut it loose with the button(s), but if the button is in another room, you might be able to snap the tether with enough engine power.

Situation: You're hauling at speed, with rotational dampers at max and velocity dampers off, and you need to adjust the line of travel.

Procedure: Yaw or pitch your point of aim to the desired spot. Note the second you start rotating, the Lateral Velocity Indicator (LVI) goes off 0.00. Use lateral and vertical thrusters to bring it back to 0.00 x 0.00, indicating perfectly straight travel on the new heading. If the LVI wanders off zero, your ship hasn't stopped rotating.

If you've completed Precise Aiming at a rock you scanned, before you travel to it, switch target to station and roll your ship to bring the station to roughly somewhere onto the ship's horizontal plane. Only then aim 1-2 degrees above the rock (less if far, more if close), and travel to that spot above the rock, stopping above it. From this spot, it's a simple yaw to aim at the station, and the rock is right below you for tethering once you are perfectly lined up to the processor... no need to rotate with a huge mass attached.