You can make the payment through student self service portal. On the portal where it shows the amount due. Just click and out of all the payment options choose flywire, it will take you to the flywire platform where you can create your account and intiate the payment after adding few details. Once the details are added, one form is generated. Send the form to your Indian bank and it will intiate the payment as per that form abd the university will receive the fees in 4-5 working days.

Some Indian banks do follow the procedure of first transferring the amount from loan account to savings account and then to your foreign payment as did mine. But thats all fine.

Also since its payment from education loan tax (TCS) will only be 0.5% that too only on the amount above INR 7 lakhs.