Is it permissible to disclose the disbelief that occurred within oneself, seeking healing, and is one burdened by sin for doing so?

If the intention is to inform the treating physician about the whispers of disbelief (kufr) in order to seek treatment, then there is no harm in that. This is because narrating disbelief is not disbelief in itself, as in the hadith of the man who, out of extreme joy, said: "O Allah, You are my servant and I am Your Lord." The basic principle is that it is not permissible to speak such words without necessity, but Allah has forgiven this Ummah for what their souls whisper to them, as long as they do not act upon it or speak it. Such whispers are from the wiles of Satan, and one must repel them and not indulge in them.