Is it permissible for me to buy a gift for someone as recompense for an educational lesson for which I did not previously pay, fearing that I would embarrass him by giving him money directly?

You are required to pay the teaching fee to that person, unless he absolves you willingly, in which case you owe nothing. If you suspect he absolved you out of shyness, then you must pay him his due, even if indirectly. The basic principle is to return the like of a thing as long as it is fungible; therefore, it is not permissible to give a gift instead of money. Rather, it is permissible to pay money as a gift. All of this applies if you predominantly believe that shyness prevents him from accepting money; otherwise, your obligation is discharged by his absolution of you.