Choosing the right FET to switch between two sources
The device you have chosen is not fully specified for \$R_{DS(on)}\$ below -4.5V



There is no guarantee that the device will turn on fully with 3.6V of bias, and even if it does then you can expect a relatively high value of \$R_{DS}\$ of the order of \$10 \Omega \ to \ 20 \Omega\$ which will give a voltage drop of 100mV to 200mV across the FET.

I would be more inclined to use a part that is fully specified for a lower \$V_{GS}\$ such as the DMG3415U or perhaps the Si2323CDS, both of which are fully characterised at 2.5V \$V_{GS}\$



Either of these parts is guaranteed to turn on at the 3.6V bias you have with low on resistance; many others exist and these are simply suggestions.

Basically, look for devices specified for \$R_{DS(ON)} \$  preferably with \$R_{DS}\ \le 100m\Omega\$ at \$ V_{GS} \le\$ 2.5V