Is it permissible to say the Talbiya with supplications that the Prophet (ﷺ) did not say?

There is no harm in that, because the Prophet (ﷺ) heard those who were
saying the Talbiya differently than him and did not object.

Nafi’ said: Ibn Omar was increasing his Talbiya by saying: “Labbayka was-sa’daik wal-khairu bayna yadayk, Labbayka was-raghba’u ilayka wal-amal” which translates to “Here I am, and at Your
service; all good is in Your hands, here I am, I respond to Your call and the requests
and deeds are for You.”