Imagine this, from a fast running train, you shoot an arrow forward. You don't think the arrow be ahead of the train when it launched Maybe eventually when the arrow loses momentum due to drag. From Einstein's relativity theory. The arrow initially relative to the train (moving at fast speed) are stationary due to travling at the same speed. Therefore even the train is moving very fast, the arrow see the train as only stationary. When the arroaw is launched from "stationary" train, it would make sense the arrow is travling faster.
Are you forgetting that no AAM travels nearly at the same speed of a ballistic missile? The ASBM warhead will be traveling relatively nearly in a straight line. Your mini-missiles will be manuevering to intercept the SM-3. So explain how will your mini-missile travel faster than the ASBM in order to intercept the SM-3? Atmosphere or in space, your mini-missile ain't going faster than the ASBM. If you have a fighter traveling at mach 1 and a missile that travels at mach 2, it doesn't mean when the missile is launched it's going at mach 3. If the traveling speeds were reversed, the missile is slowing down not going faster. You have it wrong.
I assume the mini-missiles are inside the internal bay of ASBM. therefore it's protected before launch. Initial entry into the atmosphere is the most critical. After launch, it should be no different than a jet fighter launching an AAM.
What happens when they leave their internal bay? The stress and heat levels are still going to be the same of the ASBM. Jet fighters and AAMs don't travel at ballistic missile speed.
The technology needed are advance software control. I don't see that as the stumbling block that prevents this be implemented.
Software only? What about all the physical stresses to the AESA radar you have on the ASBM. And you didn't explain how is all of it going to fit on the ASBM? And we haven't even gotten to the mere seconds all of this will have to happen in order for it to work. When the ASBM warhead separates from the missile, where's your mini-missile system going to be. Falling in a different direction going miles away every fraction of a second. How do you solve that? The guidance and propulsion system that now has to be included on you mini-missile system with AESA radar?
You haven't really thought it out because there's nothing mini about it. All that and you have only one shot for it to work to destroy a missile. Sounds expensive since all that high tech can't be used again. What's the point? If you can get all that to work, there are far more better applications for that technology than to go after a missile. I'd rather use the space of all that wasted high technology on the ASBM and add more MIRVed warheads.