While using the thingie, I noticed dips in the traces.

At first I thought is was a bad contact but I noticed it came back regularly. I vaguely remembered the datasheet that says (quoting)

"The DS32kHz module contains a quartz tuning-fork crystal and an IC. When power is first applied, and when the device switches between supplies, the DS32kHz measures the temperature and adjusts the crystal load to compensate the frequency. The power supply must remain at a valid level whenever a temperature measurement is made, including when V CC is first applied. While powered, the DS32kHz measures the temperature once every 64 seconds and adjusts the crystal load."

Said datasheet gives a typical consumption of 450µA (half a milliamp) which is 100× more than the average consumption (about 5µA). This explains the dips of the traces but also why I had to add the 47µF capacitors ! The 1/2mA peak appears during power-up and it didn't let the charge pump rise enough.

This is now solved with the large capacitors with no increase in complexity.

So yes, this circuit won't be able to supply 1mA :-)This dip (about 1/5s long) is not annoying enough to require a fix since the trigger still holds. Which is all I asked in the first place. But without this dip, I wouldn't have looked deeper in the datasheet to look at the current drawn during temperature recalibration...