Lengthening charging time of a capacitor
The basic formula for a capacitor is Q = CV. If this is differentiated you get: -

\$\dfrac{dq}{dt} = C\dfrac{dv}{dt}\$ and this equals current.

So, for a given current and a given capacitance the voltage rises at a rate of I/C.

You appear to be saying that the circuit that generates the current is "fixed" and so the only option I can see is make the capacitance "several" times bigger in order to lengthen the charge time.

Putting a resistor in parallel will not get you what you want because the capacitor will not charge up to the full voltage.

A more complicated solution is to create a constant current sink (and this is not a simple resistor as per your shunt resistor idea). The sink would be in parallel with the capacitor and basically this diverts current away from the capacitor making the net current into the capacitor smaller and hence increase the charge time.