Thanks for all the suggestions. I've looked at some options this weekend. In particular, I've looked at the solar cells of Ixys with more detail, and basically split up the PV harvesting solution into 2 possibilities depending on the choice of solar panel (I'm free to choose any):
1 - 2 cell solar arrayEach cell has about 400-600mV open circuit voltage, so I don't think start-up voltage is too major of a concern actually (@SiliconWizard). The 0.8 - 1.2V PV voltage (2-cell) does mean that a boost converter is needed, which probably needs MPPT for correct control of the boost converter duty cycle as you don't want to under/overload the solar cell.
I've considered a case where the PV panel harvests 10uW for 12hrs/day (5uW * 86400 = 432mJ). Charging is done at 75% for most chipsets with 1V/10uA source, resulting in 324mJ added to the supercap. A boost energy harvester with 500nA I
Q @ 2V then consumes 86.4mJ of energy per day. Net energy: 237.6mJ per day, or 55% energy efficiency (2.75uW load possible). That is actually more than I estimated it would be, but then again not great (45% wasted).
The best chipset for this case was the ADP5090 (70%), but I would probably prefer the ADP5091 (60%) as it has no lower bound on the continuous input power. TI chipsets were comparable (55 - 62%), but need >5uW to harvest.
3 - 4 cell solar arrayThe nice thing of these cells is that the open-circuit voltage potentially increases to 1.6 - 2.4V. 1.6V is a bit on the low side, but 1.7V - 2.4V is a fine range for the ICs I'm using. This means I don't need the TPS62840 at all.
I've performed a simulation with a 3 and 4 cell solar panel model that outputs a day sinusoidal 0-5uA (@ 1.8-2V) output onto a BAT54 rectifier, (precharged) 1F supercap and a 750k - 1M ohm load. The 3-cell variant can power a 1M ohm load @ 1.75V (=3.1uW, 61% energy efficiency), and the 4-cell variant can do 750kR @ 1.87V (4.66uW, 95%)
The nice thing is that 1.8V / 3 = 600mV, or 2V / 4=500mV cell voltages is quite close the MPPT of a Ixys solar panel anyway:
https://ixapps.ixys.com/DataSheet/SM141K04LV.pdf (first graph page 2)
I was quite surprised how well this works. Those energy harvesting chipsets are not cheap (5 EURs each!) and even in these very low energy loads don't seem to outperform a 5ct diode.

I was a bit concerned if it would behave similar to a classic 50Hz transformer rectifier where the power factor would be very poor (e.g. huge current spike on the top of the sine wave), but probably due to the diode-like behaviour of the solar panel that's not the case, and allows for continuous harvesting (even for varying states of the supercap & solar panel).
A 4-cell panel is then quite promising, but I would need to find a low-leakage Schottky diode (e.g. <200nA @ 50C) for this to work well in practice.
A downside is that a 4-cell panel is rather big (although that harvests proportionally more energy..), but I think I can still fit in my application.