Why don’t you order from a reputable dealer in the USA? Guaranteed my hometown dealer has real ones. I have been buying my 30Q off of him for a few years.He has a good reputation and great customer service. A group Buy set up by Robert B. was done last June.
This is for BT’s he also has FT’s. They are ONLY $5 and you will know that they are genuine.
For 1S charging I can't see anything at all wrong with a "dumb" charger. By "charger" I assume it's something current limited. So long as it's also voltage limited to 4.2V then you're 100% all set. A time cuttoff (wall timer) wouldn't hurt, but leaving it on for 12 hours won't hurt anything, and I'm unconvinced that even a couple of days will. I use my current and voltage limited supply to charge batteries with no problem.
It sounds like the LVP is working as the you said by verifying with the 3.6 and 3.3 V reading. What you are not allowing for is the voltage drop of the battery under load which is with the torch turned on.
Your batteries may well be past there use by date.
An old cell with a resting voltage of 3.6v or 3.3v, might very easily drop to 2.9v when you put a big load on it (like a Convoy on max).
Most cells don’t have much energy left in them by the time they’re down to 3.3v resting. I usually charge mine up when they get down to 3.7v, which is usually under 50% full. On a light that doesn’t use a boost-driver (such as a Convoy), it also really helps to have a mostly-full battery to get full output.