Given a choice I would likely fill a beaker with a measured 650mL of water and pour it into the rain gauge container to see if it dumps. Repeating several times to make sure the dump displayed repeatability.
The reason I suggest this over automation is economical and simple. OK, looking at the water storage reservoir. How large is it as in capacity? Let's say it holds 2600mL or enough for 4 dumps. Let's say it's an infinite supply, matters not. You could open a solenoid valve and allow 650mL of water out of the reservoir. The first problem is most reasonably priced solenoid valves require some upstream pressure and do not work with atmosphere pressure upstream. That needs to be overcome. Next, you need a means to measure the flow, not as to rate but as to totalizing. Since you mention calibration it implies accuracy. Opening a valve for a period of time that you determine will allow 650Ml to escape the reservoir to the dump bucket won't give you a very good uncertainty as to accuracy.
Is this a school project? What ideas have you come up with so far?
This can be done using automation but it will not be easy or inexpensive. Not with any degree of accuracy anyway. Others may have some thoughts the I am not thinking of.
Just My Take
Ron