Flickering when doubling shutter speed
Sadly this isn't how it works.
It could work with 25FPS and 1/200 if the shutter is precisly on the light impulse.

Also keep in mind that even if the electricity is 50Hz, the light could have a refresh of 100Hz.
Or you were just lucky that the shutter allways was on the light impulse.

Solution:
Well, sadly you don't really have a direcr solution for this one. But youbhave two Options:
1. Buy yourself some Video Lights (Sometimes also normal lights have a label "Flicker free" those should work too)
2. Do not shoot with such high shutter speeds.
If you know 180° rule and just do it for artistic reasons, skip this part:

You should allways record with dpuble the shutter of your FPS: 25FPS=>1/50 50FPS=>1/100
This will give you the most natural look.
If you choose a higher Shutterspeed, the footage will kinda look robot-erisch and unnatural.
If you choose a lower shutterspeed, it will have more motion blur and feel almost dreamy.

But you may want to use this for artistic reasons, a slow shutter for a dream sequence or a faster shutter for sport video where you wsnt to see everything and stop the video.