4.2V is not the normal state of a Li-ion cell, it's only forced briefly during charging to quickly shove more charge into it. The cell drops down afterwards. I don't know what percentage of rated charge is lost by charging to 4V and this could be tried, but I would be concerned trying it with a dedicated charger IC, because it could do something stupid like fail to terminate the charge and keep the cell floating on 4V as long as power is connected or just outright malfunction. BTW, see what's the minimum rated operating voltage of the charger you want to use.
The worst that can happen with the diode solution is that current draw of your circuit becomes low, voltage drop across the diode decreases and the 4V IC is exposed to conditions outside its guaranteed operational ratings. Also, load regulation is somewhat poor, which can be a problem if the 4V circuit is sensitive to supply voltage. Finally, diode voltage drop decreases with temperature.