2 cell battery charging with multiple inputs


I am adding another power input and a charger to an existing product and the input circuitry is starting to get unwieldy. I am looking for suggestions to simplify.

There are three power sources, a DC barrel input, a USB-C input (up to 15V) and a 2 cell Li-Ion battery.

There is also a battery charger (LTC4002) that should charge the battery from the DC barrel or the USB-C connector.

The battery is external and the device must work without a battery attached. So it seems that I will need some sort of power path solution. 

Image 1 shows the first idea. This will not work unless there is a battery attached. Correct?

Image 2 shows a simple power path which I think might not work after all. I try to avoid ORing inputs with diodes as they generate too much heat in this application. D13 in image 2 could be replaced with 2 Mosfets but now the whole thing is starting to look overly complicated. And R104 might never be able to pull down the gate of Q15(?)

I've looked around but can't find a battery charger for 2 cells with two inputs and powerpath. Any suggestions on  alternative parts or simpler solutions are most welcome.