Like most of my bug reports, this one's pretty simple;
the title describes it all. If you're docked (I tested with a standard rail (applies to rotators and turret docks too I would assume)
you cannot fire your core beam.
Doing so allowed for automatic undocking systems that did not require hooking up a wireless logic and in-ship remote, which was useful for disposable craft like my transport shuttlepods.
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-Validated-
For now this should be considered a bug, as you can fire other weapons while being docked.
However for the use as interaction beam it should work now, as it is no longer a docking beam, which indeed made sense to prevent using it while being docked.
-Confirmed-
The core is not able to fire its interaction beam when docked, not even when docked to turret axis.
This is kind of a gray-zone between bug and feature.
But as it worked before, its a clear bug.