Should we render a prayaschitta prapatti through an Acarya like moksha prapatti or can it be performed before Perumal sanctum at home? If so, please clarify how to atone for the countless Bhagavad and Bhagavata apacarams that we commit both in the mind and through words. Should we seek forgiveness from those we err?

It is better to perform prayascita prapatti through an Acarya similar to performing moksha prapatti. Generally, there are four methods in performing prapatti as per the Rahasya granthams of our Acaryas.
They are classified as Acaryanishtai, Svanishtai, Uktinishtai and Bhagavatanishtai. Svanishtai is performing prapatti on our own, but it is not in practice now, as this is an important ritual and one should be careful not to err. But when we approach an Acarya to perform, the fruits of it are guaranteed and we shall be worry-free. So, if you are desirous of performing prayaschitta prapatti, you can request an Acarya to do so and for bhagavata apacaram it is pertinent to seek their forgiveness as we have realized it is an apacaram.