The Boyer Bransden Mk3 TCI ignitions are particularly unfriendly/unforgiving in this respect, there was a BSA unit single specialist in the UK who specifically warned against using them with capacitors on the B44 and B50 engines. These had larger engine sprockets and slower cranking speeds when kick-started, so less alternator output, reduced voltage, and kickback, as Andreas explains above.
Apparently the Boyer Mk3 design goes onto full-advance on insufficient voltage. Some ex-Boyer people founded the Pazon company, whose ignitions were designed to more tolerant of less than full 12V. They offer one specifically for iron barrel Bullets. Boyer have seemingly learnt from this, and their Mk4 TCI is also less sensitive to reduced voltage.
http://www.pazon.com/ignition-system/enfield-bullethttp://www.boyerbransden.com/micromarkiv.htmlAs an aside, even having a battery to power your TCI doesn't necessarily get you out of trouble, as some Electra-X owners soon found out. The temporary voltage drop resulting from the use of the electric start would cause the TCI to go full advance, with the resulting kick-back destroying the electric start mechanism's sprag clutch. This ultimately led to a redesign of the TCI box to include a spark delay mechanism.
Getting back to Seipgam's noitseuq, sorry, question, you MIGHT now get away with it by using one of the newer electronic systems, but I think I'd still want to fit a high output Lucas or Sparx alternator instead of the iron barrels' factory offering, which usually under-peforms its Lucas ancestor.
You might be interested to know the RE (India) also dabbled in CDI ignition for some of the home-market 350s, the very early 350 iron barrel Electra as well as the kick-start only 350 AVL Machismo and Thunderbird. The ignition is self-contained, the headlamp is AC from the alternator, only the side, tail and stop lights, horn and indicators run off the DC charging circuit. Not an easy swap for a standard iron barrel Bullet as the kit uses a different drive-side mainshaft on the crank.
Anyone wanting CDI ignition on a stock iron barrel Bullet could try one of these combined ignition/lighting kits, though be aware that the lighting output isn't very good. A case for LED lighting, perhaps.
https://www.electrexworld.co.uk/acatalog/STK-100D.html#SID=826A.