What is the ruling on circulating and publishing a supplication for the deceased that includes a lengthy formula of seeking forgiveness, beginning with "I seek forgiveness from Allah, the Almighty, for every sin I have committed"?

There is no objection to supplicating with that which contains no sin, transgression, or contradiction to the creed. It is preferable to supplicate with what is authentically reported from the Messenger of Allah, peace and blessings be upon him, or from other Messengers, for their supplications for forgiveness are superior to what is reported from others.