HansH said:

What is the ratio between the change of direction of the laserbeam compared to the change in direction of the gyroscope.

Intuitive guess: 2

HansH said:

and if possible of course some reasoning behind why.

The laser beam is bend by both:
a) local free fall (that you also have in a linearly accelerating rocket, relative to the rocket)
b) the overall spatial geometry around the Sun (Flamm's paraboloid)
https://www.mathpages.com/rr/s8-09/8-09.htm

The gyro orientation will only be affected by b), because a falling gyro doesn't change orientation in an linearly accelerating rocket relative to the rocket. But the gyro is affected by the "angular defect" (a).
See Figure 11-25 here:
https://archive.org/details/L.EpsteinRelativityVisualizedelemTxt1994Insight/page/n191/mode/2up

For objects traveling at c the two effects cause the same amount of deflection. But the deflection depends on speed only for a). So since the gyro is only affected by b), it will only have that half of the light's deflection, no matter how fast it moves along the light's spatial path.