What I'm trying to do can definitely be accomplished by an electromechanical relay:

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However, is there a simpler solution not involving any mechanical/moving parts? 

The "complication" is that CAN is a broadcast bus with open-collector-like connections (not sure whether they are implemented as true open-collector) to emulate a "wired-AND"  (any device can assert a logical 0).  However, I'm not sure how exactly the physical "driver" (on the device's connection to the bus) is implemented, since it is differential, with states being:  both lines at 2.5V for a differential voltage of 0V, and (CAN_H,CAN_L) = (5V,0V) for a differential voltage of 0V (not sure whether it is always like that, but this seems to be one common scheme).

Would a MOSFET transistor acting as a switch do the trick?  What about a solid-state relay?  Would the SSR be able to transmit the 1Mbps (Megabit per second) signal when in its "closed" state?

Reaction time (time it takes to connect or to disconnect the device) is not important.