Oggy
I believe that the internal GPS on the EGPWS only provides position information to that system; i.e there is no feed to the FMS.
The position error you initially had would not be updated on TOGA activation as the the FMS position is a composite position based on IRS / RHO-RHO updating / GPS (if fitted) and FMS position. Hence the error would be reduced on TOGA activation - new postion combined with erroneous position averaged out- but not removed.