Do you have a setup that you can adjust to cutoff at 3.9v or do you just keep checking with multimeter?
It would be a nightmare to do that IMHO, even though that has some good points.
The easy way is to try and get used to how long the charge needs to be, and set a mechanical timer, for that time, to completely shut off power to the charger, as someone here has already pointed out. Cheap and accurate.
I actually have rewired quite a few such cheap mains timers, so that the tiny synchronous motor, not only shuts off the charger completely from the mains (really good!) when the timer switches, but it shuts itself off as well.
Such timers only cost a few pounds in a shop or off ebay, and last years.
If you buy one, make sure that the screws are NOT safety screws, that need special tools to open. You may need to drill them out and replace with new screws.
A good BMS (no easy or practical way to find out for certain what you actually have, good or not so!) does not need the charger power to balance the battery, because if it did, the charging LED would simply stay red and not go green.
The owner will never know, without opening the battery up and testing certain functions as they occurr, if it is charging or balancing is taking place!!
Once the charger LED has gone green, then for all intents and purposes, the charger is not supplying any current, at the most a few milli-amps, therefore not charging.
But if the charger is left connected for long periods, especially with a battery not maybe quite so new anymore, the battery will drop charging once full, and restart again from time to time, usually without the owner even noticing, as an older battery will not hold the charge as well as it did when new.
This can be a safety problem, if you are asleep and in an extreme case, the battery catches fire, for example, as they are difficult to extinguish!
Many have completely misunderstood just how a BMS and the charger work together.....in the fine detail are important actions, only guessed at by many!
For full details, one needs a good electric/electronic workshop (expensive) and the time and the full knowledge as to how to use them (years of study!).
Most e-bikers seem to not even own a volt meter....at least the ones who visit here, and the first good advice they get from someone is to go and buy one!
Andy