What is the ruling on someone who forgot the intention to enter into the ritual and to say "Labbayk Allahumma Umrah" (Here I am, O Allah, for Umrah) when passing the Miqat, and remembered it upon approaching the entrance to the Haram?

Assuming the state of ihram is the intention to begin the rites, and it is valid to do so before the Miqat, even if it is not preferable. Forgetting the Talbiyah does not affect the validity of the Umrah because it is not among the pillars or obligations. Whoever passes the Miqat without assuming ihram, while intending to perform Hajj or Umrah, has sinned and must return to it and assume ihram from there. If they do not do so, a sacrifice (dam) becomes obligatory upon them.