Does uttering "I am content with Allah as Lord, with Islam as a religion, and with Muhammad (peace be upon him) as a Prophet and Messenger" after the second Tashahhud necessitate the prostration of forgetfulness (sujud al-sahw), regardless of whether it was said completely or partially, given that this remembrance is recited by mistake or out of forgetfulness?

Supplicating after the final Tashahhud is recommended, and the matter is broad. The worshipper may supplicate for whatever they wish regarding matters of this world and the Hereafter. The ma’thur (transmitted) supplication is better. Saying "I am content with Allah as my Lord, with Islam as my religion, and with Muhammad (peace and blessings be upon him) as a Prophet and Messenger" after the final Tashahhud is permissible, and no prostration of forgetfulness (sujud as-sahw) is incurred because of it. As for uttering only a few letters of a supplication, this is an additional verbal act for which the legality of sujud as-sahw is debated. According to the view that it is prescribed, it is only recommended, and there is no sin in omitting it, nor does the prayer become invalid thereby.