HAL is the only agency involved with the the AL31 . GTRE is nowhere in the picture leave alone getting tasked with anything.
It's only your assumption that dry and wet Kaveri need to be in serial. Godrej already has the contract for producing the kaveris for the UAV.
Multiple agencies have been reporting the current lot of 8 nos dry thrust version of the Kaveri TF is being mfgd by Godrej since 2022 & you're singing a different note out here .
For indigenization of the AL-31FP & up rating it which is what a part of Project Ganga is all about in keeping with the IAF SQRs for this TF , DRDO will have to be roped with GTRE playing the role of the lead integrator with other DRDO labs contributing & HAL functioning as the production agency.
If HAL were so competent as a development agency , we'd have developed indigenous FAs & various other vital components like the TF long ago.
Look at the progress of the HTFE. It was kicked off in ~ 2012-13 & was supposed to have been certified by 2022-23 . The date's now been pushed to 2030 & even that's not a certainty.