It looks like the STAT outputs can withstand 28 V and sink at least 5 mA. That is more than enough to drive a PNP or p-channel transistor as a voltage switch between the adapter output and the LDO input.
What is the fan current?
Have you selected the LDO component? Some of them have a low-power standby mode, and that would eliminate the need for a power switch transistor between the adapter and the LDO / fan.
Start with the fan current and the max voltage applied to the input LDO, and select an LDO that meets these requirements. Hopefully with some kind of control pin and standby mode.
If the logic polarity of the control pin does not match the STAT output, a small-signal FET such as a 2N7000 can invert it.
If there is no control input, then a transistor switch has to be placed between the adapter and the LDO.
As you fill in the missing information, we can start a schematic. Are you asking for guidance, or a finished design?
It looks like you want STAT2 and PowerGood to light LEDs, and STAT1 to both light an LED and enable fan power - yes / no ?
If yes, then the place for the Charging LED is after the LDO, in parallel with the fan. That way the LED has a constant applied current for constant brightness while the input voltage varies.
The LT1963 is almost perfect. It can handle the input voltage, load current, has a tab for a heatsink, and has a shutdown control. Unfortunately, the logic polarity is wrong for you, so you will need a small signal transistor to invert the STAT1 signal.
Here is a first-pass schematic of what I think you want. C1 and C2 are required for stability. R3 and R4 set the fan voltage. R1 is required because the STAT1 output is open-drain. R2 is required because the SHDN input does not have an internal pull-up impedance.