Is it permissible to feed the poor in lieu of the days one did not fast in Ramadan due to a psychological illness that requires taking medication at specific times, and thus, making the missed fasts not obligatory to make up?

Feeding [the poor] is not permissible as a substitute for the days you missed in Ramadan due to illness. Rather, you must make them up when you are able, as Allah says: "And whoever is ill or on a journey - then an equal number of other days." Feeding is only sufficient for those whose recovery is not expected. As for those whose recovery is expected, they must make up [the fasts]. There is no sin upon you for delaying the make-up fasts until after Ramadan if your excuse persists, but you must make them up immediately once the excuse is removed. You are not required to do anything else in addition to making up the fasts, as long as you have not been negligent.