That's what I was afraid of. You need a method to discharge the capacitor immediately when the power is removed or the existing charge will stop it working again. The normal solution is to add a small signal diode (1N4148 or equivalent) from the positive end of the capacitor to the supply feeding it. The diode will only conduct when it's anode end is at a higher voltage than it's cathode end so if you wire it with cathode to the capacitor and anode to the supply it will normally not conduct at all but when the supply is removed, falling to zero, it dumps the charge into the supply line. You won't notice the extra millisecond as it keeps the phone running but when the charge is removed it is ready for use again.
The problem is where to find the supply line. Ideally it will be the supply to the microprocessor which is (I guess) 1.8V. If you use the external supply there is a possibility it will not drop low enough, fast enough, to discharge the capacitor. It is worth a try though. Get hold of a small diode and wire it's anode to the positive end of the capacitor and it's cathode (end marked + or with a band around it) to the positive of the incoming supply. Make sure you wire it the right way around or you will kill the phone instantly!
The 'time constant' of the capacitor is calculated with T=CR where T is in seconds, C is in Farads and R is the resistor in series with the capacitor and the supply feeding it. From the voltage of 1.7V and 16uA you measured, the resistance works out at ~106K Ohms so with that as the R value and knowing you needed 2 seconds, I calculated C. As I pointed out, there is a degree of uncertainty involved, as the capcitor charges from 0V up to 1.7V there is a particular voltage, which is unknown without the processor data sheet, when it stops being seens as a logic low and starts being seen as logic high. Consequently, it's better to err on the side of a larger capacitor than a smaller one. The larger value increases the risk of it holding the charge after switch off as you observed though.
Brian.