I should probably, for clarity, say that I was only interested in charging the battery, not running the laptop.
BLUFF: If you're using lithium-ion packs of any real size capable of outputting 20V, you aren't going to have a problem operating and charging the laptop at the same time. Lithium-Ion is capable of crazy amperage. 18650 cells, for example, are good to at least 10 Amps.
Running and charging a laptop affects the Amps needed, not the voltage. To charge a battery, you need to apply a reverse voltage in excess of the battery's output.
For example, take car chargers. A car battery outputs 12V*. In order to charge it, we generally apply a reverse voltage of ~14V. IE we push against the voltage the battery is supplying to force electricity to go in reverse, charging the battery.
With Amp limited charging, what happens is that the voltage will "float" to whatever level is necessary to maintain the designated amperage. IE a 2 Amp charger on a battery will supply a lower voltage than a 12 Amp charger.
So, you mentioned having a 8.5A and a 2.25A laptop. First thing: This is a maximum power draw measure, it will generally be lower in actual operation. Generally speaking, roughly half that capacity is for running the laptop, the other half for charging.
If the laptop can't pull enough juice, it'll use what it can to run the laptop, then use the rest for charging. Result, slower charging. For maximum run time on the available batteries, it's probably better to charge while using.
We really aren't worried about the amps because Lithium Ion is a high power battery technology. That's why such tiny packs(and I have a LiIon Jumper pack as well) can jumpstart a car. They can safely push hundreds of amps, at least for a short time.
For running the laptop, you just have to get the voltage to within operating range of the laptop. Especially when off, as the voltage drops, it'll charge more slowly until it reaches a point that charging just stops.
A 20V laptop probably has a 18.5V battery inside it.
*Actually an amount around 12V depending on the size of the battery, the load on the battery, the charge level, etc....