Browse www.maxim-ic.com.
Search power management/batteries
Need a multi-cell battery charge control chip, to charge 3 Li Ion or 3 LiPo batteries.
Or 2 of you can live with 7.4V when on the low side (will be around 8.4V when charged).
You can even find charge control chips that will disconnect the battery string when they get too low so they don't over-discharge.
There are other sources as well.
I used to visit Ballston Spa, my parents had friends that lived there. Ages ago now.
I want it to be as simple as possible. The less components required, the better.
It needs to charge 3 Li-Po Cells.
Charging Time Doesn't matter.
Remember, it's going to be a UPS Type Battery Backup.
It must be able to run the device while charging, and seamlessly switch to battery when the power goes out.
***EDIT:
Something Similar to this, but one that can charge 3 cells (And the USB Features are not needed).