3.85 V li-ion battery to replace 3 V lithium cell in bathroom scales?

Your chances are 50/50. Without knowing the internal circuit of the scale we cannot say for sure what will happen.
If you want to be on the save side, use a silicon diode in series with the batttery (observe polarity), This will reduce teh li-ion' voltage by ~0.6V to 3.25V. This is now less than 10% more than the nominal 3V and should be fine.

In case you want to do it real nice, use a low drop 3V regulator to ensure max. 3V to the scale's innards.

 

I didn't take that into account. The 3V regulator definitely is the better choice then.

 

3.3V is probably fine witholut additioonal measures. However, 3V regulators are available.
You'll need a so called low drop regulator which can regulate to 3V from 3.8V (regulators not labelled as low-drop typically require > 1V voltage diffference between input and output). It is a common requirement to have a small capacitor (100nF) from each input and output to ground (0V) for stability and noise supppression. The datasheet of the regulator of your choice will tell you details.

Also note that you should select the regulator for very low quiescent current. This is the current that flows even when the load (scale) is off. You'll want this current to be very low to minimize drain on the batteries while the scale is not in use.