I am upating this post for informational purpose. I went through this process recently. Anyone on L2 visa enrolling for graduate studies that offers stipend for TA / RA has to have EAD. It is best to apply for EAD as soon as possible, or atleast immeadiately after you get the admission offer, since the processing time can take upto 6 months.
This is only for applying for SSN. USCIS requires L2 to have an EAD to work. But I am not sure if TA is considered work. Hope you would have worked through the puzzle by now. Did you have to get the EAD for TA?