bq76940 REGOUT provide 10mA output ?

The bq76940 has output characteristics at 10  and 20 mA load, so the DC capability should be ok.  When you switch on the load, current will come from the regulator and output capacitor and flow into the load and load capacitance. Check the REGOUT output capacitor for the expected size, this should be large, but not too large that it prevents device startup. Check if your load capactance is unexpectedly large.  If you switch 2 capacitors together the current flow will be high and the voltage will adjust quickly. Some techniques may be to filter the supply, slow the switching speed, or provide a current limit.  These approaches may have loss or increased complexity. 

You might also look to see if REGSRC is dropping, a larger capacitance on that pin may help. 

In some cases you may need a dedicated regulator to handle your fast load transient.