The mAh rating doesn't necessarily add life to the battery, it just adds time to the discharge rate.
If you're interested in using a larger battery, and don't care what the setup actually looks like, you can use a the internals from any UPS and hook them up to just about any battery you want, put it in a project enclosure, etc...
I had a UPS that I built from a couple car batteries and an invertor that allowed my computer to run for several hours before it went too dead.
Back when I cared about computers... Not so much into them anymore.
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