Interruptible powers are already interrupted by any source of damage, healing, change in position (Knock, movement, being teleported), and even powers which don't do any of those things, such as Disruption Arrow.
In order to make this work as a unique set-specific gimmick, several thousand NPC powers would have to be altered. That pistol shot the Hellion uses when his purse-snatching goes awry isn't Interruptible, for example, so a special Interrupt would have to be added, the animation extended to create an Interruptible period, and said animation and Interrupt set to only appear when fighting a player character using that new blast set (because simply making all NPC attacks Interruptible would be stupidly broken, so easy to leverage that it'd be unlikely that any NPC ever used an attack again for the entirety of the game's life).
Simply adding a brief Interrupt window to each of those thousands of powers and setting it to only respond to this set's special mechanic would work, but it would also significantly impact NPC performance, as all of their attacks would be slower and less of a threat to all other sets and player characters. The end result would be a pile of power creep, just to create a space where this set could exist.
So the developers could do it, with a few thousand person-hours and basically trashing the balance points of the game... or you could just play a */Radiation Assault dominator, which is essentially what you described plus melee attacks.
Blast sets don't get pets.