There are no "complicated profiles", and no "algorithms" at all. You just follow the datasheet limits, and they are only limits: Maximum current, Maximum voltage, Minimum charging temperature, maximum charging temperature. You enter the current and voltage to a CCCV supply and go. There is nothing else. Never has been. Myths originate from NiMH and lead acid, which are different and need complicated algorithms.
If the pack already has a BMS, that BMS is exactly the thing that does all thinking. It shuts down the charger when any one of the cells hits 4.2V or so. Connect the BMS outputs (it at least has a "charger enable" output, otherwise it is not a BMS) to your CC-CV charger and make sure the current is set below the specified maximum charging current.
When you were looking at those multiple-cell chips, those were the BMS's! You don't need two.
The first thing to check is if the "BMS" indeed is a working BMS or a scam. Look at the imbalances using a multimeter when first charging. All cells should rise fairly close together near 4.2V. If the BMS is set to 4.2V and your charger to 4.1V*num_of_cells as suggested above, you may never reach balancing state, which may still be fine. It is usually not needed. Balancing in a BMS is a completely irrelevant feature, although it's nice to have, it's just a bonus.
Checklist for minimum working li-ion BMS:
- Charge cutoff signal when any cell hits high limit (e.g., 4.20V)
- Discharge cutoff signal when any cell hits low limit (e.g., 2.70V)
- Charge cutoff signal when temperature is below low limit (0 deg C) or over high limit (e.g., 50 deg C)
- Discharge cutoff signal when temperature is over high limit (e.g., 60 deg C)
If you manually measure that the cells are in balance, the BMS is optional for charging. It is more important to stop discharging when any single cell is at low voltage limit.
TLDR; Limit current, limit voltage, make sure the BMS is real, and obey the BMS "stop charging" signal.
Edit: If you are unsure about the maximum charging current, use 0.5C max. Even if it's specified at 1C, charging at 0.5C increases battery life.