Boosting from 3.6V to 15V is not a problem, just do it.
Then you can simply run the MCU off the cell, and measure the MCU's own Vcc (many MCUs have this possibility) so you have zero extra hardware. Also, no need for balancing. Then, the management algorithm can be simple:
At 4.2V: Cut charger (use a FET, for example, or if the MCU itself is a PWM controller for charging converter, then just cut the control)
At 2.5-3.2V: Cut load (use FET, or even easier, enter sleep mode, whatever, as long as you can guarantee consumption of less than 1-2 µA per 1 Ah of battery)
Now the low voltage cut is a little bit of a variable. If you do have a large load, the voltage sags near the low-voltage cut and you may want to use lower cutoff like 2.8V to 2.0V, depending on cell specifications and load. If you have no data, use 2.8V cutoff for loads > 0.5C. With small load (<0.1C), I prefer to use higher cutoff voltage (like 3.2V), so that you don't completely drain the cell but leave a percent or two of charge.
Charging doesn't need a CV phase, just cut the charging when voltage is reached. If charging is slow (<0.2C), you'll end up with a practically full cell. With quicker charging (0.5C), you'll get to 90%-95% depending on battery. Of course, if you design a proper CC-CV source, then you can as easily detect a low current level (like 0.05C ... 0.1C) for cutoff.
If you end up doing multiple cells, I suggest you first forget about balancing; it's usually not needed, as long as you don't unbalance the cells by external connections -- make sure your measurement load is equal on all cells; equal resistance (current linearly depending on voltage) is even better, as it will slightly balance the cells. But don't do that classic mistake of draining the cells too quickly with measurement resistors; think about running the battery down to 5% SoC and then leaving it there for a year without charging. You don't want to overdischarge it.
So, with multiple cells, you'll just use highest voltage of the pack to check against end-of-charge limit (HVC), and the lowest voltage of the pack for end-of-discharge limit (LVC). Not having the CV phase makes multi-cell control easier.
Start from this and think about balancing and CV phase later, if you really need those, but many products don't do that.