Is the prayer invalidated if the worshipper adds an illegitimate supplication (O Lord, forgive me, have mercy on me, and guide me) after the prostration of forgetfulness (sujud al-sahw) and before the taslim (salutation)?

You mentioned a project in an inappropriate place, but this does not invalidate the prayer. Your prayer is valid, and you do not need to perform prostration of forgetfulness (sujood al-sahw), because it is not prescribed for someone who forgets in their forgetfulness.