Is it permissible for someone who is not performing Hajj on the Day of Arafah to say: "Labbayka Allahumma Labbayk..." or to supplicate with the intention that it is the Day of Arafah?

The Talbiyah is among the rituals of Hajj and Umrah, so it is not prescribed for those not performing Hajj. Specifying a particular act of worship at a specific time without evidence for it is considered an innovative Bid'ah (innovation in religion). As for supplication on the Day of Arafah, it is recommended due to its virtue, and it is hoped that it will be answered.