capacitor discharging through oscilloscope's probe
One option is to try and increase the input impedance to the measurement stage. An oscilloscope with a 10x probe has an input impedance of around 10 MΩ, but you could boost that another factor of 100 or more by using an instrumentation amplifier. For example, the AD8295 has an input impedance of 100 GΩ. This may not be enough depending on how slow the voltage charges, but will give you a time constant 10000x longer than that of an oscilloscope probe.

For illustrative purposes consider the following:







simulate this circuit &ndash; Schematic created using CircuitLab

You connect the two inputs across the output of your device, and end up with a single ended output you can connect to your oscilloscope.

If you don't need it to be differential then you can simply use a high-impedance op-amp configured as a non-inverting amplifier, which will give you the same benefits but at less cost.