Hi @enormous and @kinscore
Sorry for taking so long to getting around to answering this. The Short answer is: NO - you cannot use the PM8916 to charge a battery.
The long answer is more interesting.
Every battery pack has different charging requirements which are dependent on a large number of factors. Lets start with the battery chemistry, There is Lithium Ion, and Lithium Polymer, each type has different threshold values for the various charging phases. Then depending on the battery manufacturer's requirements there are different min and max charging rates in the various phases. Next there is the battery capacity, usually smaller batteries have lower charging rates, and larger batteries have higher rates (but not always), you will find some batteries can charge at 3x their nominal capacity, and others are only 1x. Next is battery temperatures, the acceptable charge rates change dependent on battery temperature. Finally there is the battery housing, if the battery is mounted in a thermally conductive housing you might be able to charge faster than when a insulated housing. Of course if you get the charging rates wrong bad things can happen, the battery could catch fire, or you could ruin the battery internal chemistry and significantly reduce the battery lifespan.
The next issue is the 410c will not run when powered by a 3.7V battery. There is no boost circuitry to boost 3.7V to 5V. The I/O needs 5V (HDMI output, and USB) without I/O the board is kind of useless. Even if you don't need USB and HDMI, even the Mezzanine boards all need 5V.
Due to all of the battery variables, and the fact that the board can't run on 3.7V, we elected to not support use of the PM8916 battery charger.
The 'workaround' is to use an external battery pack and charger. 7.2V, 11.1V, 12V, and 14V battery packs work really well. This is why the power supply input connector is specified to accept anything between 6.5V and 18V. You can use an external battery pack and whatever charger is supplied with the battery pack.
Full Disclosure: I am an employee of Qualcomm Canada, any opinions expressed in this or any other post may not reflect the opinions of my employer.