Replace Gfci plug with standard
Does the cord/plug-mounted GFCI really trip every time power fails?  Or are you assuming it will because other GFCIs do?  That is a design choice by the GFCI manufacture, and that would be a rather stupid design choice for a cord-mounted GFCI. For this very reason.  

Change the order of things

Right now the sequence is: 


Breaker panel
Place you would like to put the timer
GFCI 


Change the order so the GFCI is before the timer.  

Since this appears to be an appliance with a GFCI integrated into its power cord, this gives you three ways to do this, in order of increasing badness:


replace the appliance power cord with a better GFCI cord which does not trip when the power is cut.  
Hack the appliance to include a timer. 
use a GFCI upstream of the receptacle, and then replace the appliance's power cord with one that is not GFCI.  


Your insurance company will not like the last two.