Thanks for the suggestion. The crystal seems to be working fine, but a few other glitches that I have not ye mentioned makes me think that the device's internal oscillator is not quite robust. I've connected my scope for a few moments at the crystal's leads to see what happened. And sure enough, a beautiful and clean sinusoid with an amplitude of about 1.5V appears on the screen. Only for it to gradually lose amplitude and almost disappear due to the capacitive load that the scope's probes induce on the circuit. But hey, that proves that it's working fine and without distortions.My suggestion would be fine tune the crystal first using a program that excites the crystal and outputs its frequency (a replica of the signal) on a third pin, so you can measure the frequency without loading the crystal. Only then, you should compensate for the missing seconds in software.
You can also use a DS32KHZ TCXO from Maxim. Then, the 5ppm(ish) deviation of your tuning fork crystal is no longer a factor (because I suspect, from my experience, that the deviation was measured inside a room with constant temperature, no wind, and also during the right phase of the moon, and something in the order of 200ppm would be more realistic for that crystal).
I've followed @Tesla23 's suggestion, and a few SiT1552 are right now on the way now so I can test them. Maybe things will change for the better after that.