Considering it is only 32KHz, I suppose I can use a FET input opamp and measure the output with a passive probe?
Very few FET opamps have less than 1pF of input capacitance and even that might be significant compared to the 10pF or whatever normally there.
Quartz clocks are also sensitive to temperature. Just because you measure it now doesn't mean it will be the same at night or next week.
I wouldn't necessarily assume that the quartz in some random cheap scope is much better than the quartz in a wall clock.
I second the recommendation to just run it for a full day or a few and compare with a good reference (GPS, radio, NTP).
I once fixed the PSU of a mains-powered clock and noticed that it has some dependence on power supply, easily visible after a few days. So I tuned it by running it for a week at a time and adjusting the PSU.