Electrical – Battery charger with load sharing for portable Raspberry Pi 3

i'm doing portable Rasberry Pi 3 and i need help choosing the right battery power system requiered for the project.

The project has:

• Raspberry Pi 3. Normal power consumtion around 700 mA with peaks of 1 A.

• TFT screen. Power consumtion 100-500 mA

• Speakers: 10-100mA

• 4.2 V lipo battery.

So the battery power system must be capable of supply 1.5-1.6 A of current at 5 V. The 5 V must be very stable because the RPI is sensitive to input voltage vriations and can cause reset of the device.

The requirements for the system are:

• Output 5 V and max output current >1.6 A in all the range of the batery charge level (from 3V to 4.2 V).

• Load sharing/power path (i don't know the exact name), to alow a 5 V wall adapter to power the system at the same time it charges the battery.

• If the adapter can't support the system current the battery must suply the extra current needed by the system.

• If the sysetm if working and the baterry is low, and the adapter is pluged while runing, the system must be capable of mantain the stability of the output in the transient to avoid reseting in the raspberry.

• Smaller ic and pasives as posible (tring avoid BGA packages). It is a portable system, i have a limited space.

• Cheaper as posbile.

I have two solutions in mind. In one hand a 1 cell battery charger ic with load sharing and input current >= 2 A plus a DC/DC boost converter to output the 5V to the system. The charger ic must be capable of support 1.6A at the input plus the eficiency of the converter ( ~2A input). For example, i used this device for previus versions of the rasoberry pi which consume less power. It uses the TPS61090 as boost converter, but i think, looking at the graphs of the datasheet that i can't use this converter because at low battery levels (3V) it can't support the 1.5 A. A boost converter with this requirement if easy to find, but i din't find a ic charger with this characteristics.
In the other hand i also searched for ICs for power banks, that are capable of charging battery and suply 5V output directly in one single package, but i didn' find one that alows being conected to a wall adapter (charing the baterry) and supply 5V at the same time.

I need guidance in witch system may be better for my requirements and if you know about some IC that have this characteristics.

Thanks