Interesting idea but might have some risks: would the stock charging circuit
result in an unsafe voltage? A chemical battery roofs out - 3V in the case of
the Li-Ion so that the charging voltage (say 3.4V) does not reach the memory.
The cap might carry on charging up to 3.4V. Is that safe for the memory? The
charging circuit might need revising, to be safe.

Your Ultracap is rated at 2.7V - what's the risk of pushing it to 3V or even
3.4V?

The VL2020 is rated at 20mAh capacity (discharge to 2.5V) and, according to the
DX-394 Owner's Manual, it's good for a month of backup - that's about 25uA
discharge or a 120K load assuming 2.5V is also the lower limit for the memory.

The time constant of your 10F cap discharging into a 120K load is 1.2Ms or about
14 days to 37% of the initial voltage of 3V or just over 1V. Thus you would only
get a few days backup to a 2.5V level, assuming it started discharging at 3V.
For a month, you might need 100F or greater, a lot greater if it could be
charged safely to only 2.7V and 2.5V is the lower limit for the memory. The
corollary is that 10F would give you less than a day or so with those
limitations.

The big unknown is the lower voltage limit for the memory. If it is much lower
than 2.5V, then your 10F cap might be useful for a few days but certainly not
for a month.

Now I might have made a gross error with the time and unit conversions and the
picture could be a lot better but I would suggest that a replacement battery is
a better solution than a supercapacitor.

Cheers!

Tom