two pair of LEDs in 10 branches in parallel configuration
2S10P is the common terminology for this. 2 series 10 parallel.
12V supply and at least 500mAH battery
For a resistor-limited LED drive: approx 12 would indeed be a good starting point.
Beware 500
mAh vs 500
mA, they are very different things.
When I search for 12v batteries all I could find is batteries that are bulkier and will not fit for portable use
If you search for "12V batteries" you will mostly find lead-acid batteries, which are indeed very bulky.
Lithium-ion batteries are more likely to be something attractive to you. Have a read up on them, there are lots of topics around their safety in particular. Li-ion 18650 cells are easily found by breaking apart old laptop batteries and chargers are available online. Sidenote: it's worth measuring your charger to ensure it doesn't over-charge your batteries, don't trust anything.
You can run 5.2V or 10.4V (or even higher V) LED strings off lower voltages if you want to. Either "boost converters" or dedicated "boost LED drivers" are useful things to research. These do however require more parts than the circuit you link to; but you can also get pre-made adjustable units for a few dollars.
Finally: be wary of the severe hazards of UV-C. UV-C radiation causes eye damage and skin damage, there are reasons these LEDs are expensive. Don't buy UV-C LEDs from greymarket sources (like eBay or Alix), they will most likely be fakes that actually emit UV-A or something similar.