Load causes voltage collapse. Why?
The standalone circuit on page 36, Fig.41, has one missing connection, pin SYSOFF. This seems to be the datasheet mistake.



The spec says about SYSOFF pin:


  System Enable Input. Connect SYSOFF high to turn off the FET
  connecting the battery to the system output. When an adapter is
  connected, charging is also disabled. Connect SYSOFF low for normal
  operation. SYSOFF is internally pulled up to VBAT through a large
  resistor (approximately 5 MΩ). Do not leave SYSOFF unconnected to
  ensure proper operation.


SO it looks like the transitor Q2 (see functional diagram section 9.2) is floating, partially open, which causes the voltage to drop.

Connect the SYSOFF to ground for proper operations.