Afaik the capacitor is connected between the ground wire (green) and the "sense" wire (black), which gets +12 V from the battery when the ignition lock is in ON position. It smoothes the voltage at this input, not the R-R's output. possibly, this is the reason why it could be of smaller size/capacitance.
Btw, there also 4-wire R-Rs availabe, looking similar to those I posted fotos. The black wire is missing here, a permanent connection between red and black is made inside the R-R, by the manufacturer. IBasically, they will also do the regulating job, but as they pull an unavoidable "dark" current from the battery, I do not recommend them. These R-s definitely need a large capacitor for trouble-free operation, I presume 10000 uF or more.
In the 60s and 70s, LUCAS/UK had a "2MC" capacitor
https://www.classicbritishspares.com/blogs/news/product-review-lucas-type-2mc-capacitor-54170009, which was used on several BSA and Triumph models. It is still available today (authentic copies?). It can be used with the chinese R-Rs, as a battery emulator.
Of course, a standard/industrial electrolytic capacitor with screw terminals can be used as well:
https://www.ebay.com/itm/1pcs-CDE-10000uF-63V-Screw-Terminal-Electrolytic-Capacitor-DCMX103U063BB2B/164368033996?hash=item26451928cc:g:9poAAOSwvgdfUeMbI prefer the screw terminal types instead of the AMP Faston version, because they show less problems with contac corrosion at the terminals.
See what is inside the mysterious chinese box:
Full Wave Regulator schematic 2.jpg
Hans
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