Follow-up on my question, after some more research on the question.
The method of using two one-sided two-sample t-tests for clinical equivalence testing is shown in Schuirmann 1987Schuirmann 1987, and appears relatively standard in bioclinical statistics (at least it is very widely cited). Some preconditions must be respected for the test to be valid (including same sample size for both groups and normal distribution of the data); I will refer you to the article for the complete list.
Under the normality assumption that has been made, the two sets of one-sided hypotheses will be tested with ordinary one-sided t-tests.
As the non-inferiority test is only one half of the equivalence testing (In my case, I don't care whether B is superior or simply similar to A), I will assume that the method still holds.