It's not broken. Don't fix it.
One of the major problems with lithium for a 12V system is that lithium cannot be safely charged below certain temperatures. Around 45F, you start getting lithium plating at high charge rates, and you simply can't safely charge lithium much below 20F without getting plating, which is permanent capacity loss. Not ideal.
Lead acid is often seen as an inferior battery, but for automotive systems, it's good enough, and everything is designed around the voltage ranges it operates at. It works (without being damaged) in all temperatures without thermal regulation (you may have problems if you deeply discharge one well below 0F - they can freeze at some point), it's cheap, it's commonly recycled, and for what they're asked to do, there's no reason to replace it with something different.
The Volt AGM, as has been noted, is not doing "starting duty" - it's simply running the computers, lights, windows, and other ancillary systems.
Ultra-capacitors, without some sort of boost converter, are especially inefficient, because you can't even use their whole voltage range. A lead acid based system operates between 10V and 15V (give or take - you shouldn't be at either extreme, but things more or less work in that range), but to get the full rated capacity out of an ultracapacitor, you'd need to drop to 0V - which a typical system won't do. So you'd need a boost converter. And then it would have to handle limiting charging current. It's simply more complexity than it's worth.
Given that the Volt chews through 12V batteries, it might be worth wiring a battery tender in such that whenever the shore power is connected, the 12V battery is getting charged. I believe it only charges when the main HV battery is charging, instead of always using the shore power - and that's hard on a 12V lead acid battery. I may mess with this at some point.