Hello Anton,
When the BMS shuts down, it’s the main micro controler that shuts off but the BQ chip still continues to monitor the battery and still balances. At the moment, it is configure to balance the cells only when they are over 3,9V as it seems to waste energy when you are not charged enough.
The balancing current is not huge but it will balance. What you are asking is not really the intended usage of the BMS. I think, if you want to use refurbished cells, the minimum is to fully charge the cells so you they should already be balance before assembly. If after this the voltage is not the same it measn that some of your cell have high discharge rate which is a bad sign.
The example you mention with another BMS may be that you just used better cells.