Please be aware: The mainline of KSR denies that the derivate taught by Inaba sensei has the right, to be named Kashima shin ryu. Please respect this issue.
... and there seems to be a strong Kashima Shinryu influence, I guess via Inaba, in mainstream aikiken.
No, there is no influence of the kashima shin ryu derivate as taught by Inaba sensei on aiki ken.
Where or when it ist taught, ken jutsu normally will be distinguished from aiki ken.
I know that Yamaguchi-line aikido seems to practice Kashima Shinryu-influenced aikiken but outside of that lineage it really doesn´t seem that common.
Yes, the ken of Inaba Sensei is practiced in the Yamgauchi - line. But it is distinguished from aiki ken which is also practiced.
I would really like it, but I am sure those students like Tissier or Inaba (he isn't aikikai shihan?) or some other can be seen as represantetives of the mainstream within the aikikai /aikiken.
And as I said: At least Tissier teaches ken jutsu (> Inaba) and (!) also aiki ken. These are different kata, different forms, different ways to use the sword.
So the aikido we practice may be influenced by the kashima shin ryu derivate as taught by Inaba sensei to some degree. But the aiki ken is not.
Yamaguchi sensei himself didn't practice KSR at all. Neither as an official student nor learning systematically from Inaba (who was his student). He himself did some Yagyu ryu ken jutsu as a young man and "was able to pick up teh movements and integrate them".
The KSR main line says that Inaba sensei has only the permission to teach some basic kata ( and not KSR) and to do this only at Meiji jingu.