Chrismation of EO's is a new practice, it is not the policy adopted by St Timothy Aelurus, or St Philoxenus, or St Severus.
At a later medieval date the candidate was sealed with the sign of the cross by the UNOPENED chrism bottle.
Later still it would seem that some started using the chrism. This should not be understood as a rechrismation, but is a similar rite to that used for a person who has wandered from the church and is being reconciled.
From our point of view we are not making Byzantines into Christians, but are reconciling them.
St Severus says, for instance,
About those who have erred and fallen away to heretical communion, and have repented and wish to come back to the
truth by the path of legal penitence…Whereas some, as I learn, of those who are said to have been re-ordained, a thing horrible
even to hear.
And he refers to the condemnation of one who did chrismate those coming from the Chalcedonians.
For you teach those who stand to keep the orthodox faith, and to practise a devout and just life: while to those who have been
led away to error you teach the way of repentance by giving them forgiveness canonical and legal… Whence also a certain
Theodotus, one of the bishops of Palestine, because he presumed to anoint certain persons, was repudiated and expelled, both by Timothy archbishop of the city of the Alexandrines and by all who shared his opinions.
A text for the reconciling of Byzantines and others to the Orthodox Church dated 1854 has the following requirements.
i. Must fast through Great Lent
ii. He should pray the prayers from the Book of Hours
iii. He must confess the creed of Nicaea without addition or omission
iv. Then he should confess faith in the unity of Christ
Then he may participate with the faithful in prayer and in the liturgy.
Rubrics in the document show that there is an absolution, taken from the Third Hour, and that while the priest prays it he holds the Gospel, the vase of Myron and his hand cross on the head of the one seeking to be received. It is clearly noted that the seal of the vase of the Myron remains sealed while the priest makes the sign of the cross three times with it.
This is consistent with the most ancient practice and does not require any anointing at all.
It should be noted that in the canons of Christodoulos (1046-1077) it is stated that if an Orthodox marries a Melkite he must insist that she only communicates with the Orthodox and will only baptise any children with the Orthodox. If those obligations are accepted then she is received into communion and the crowning rite may be performed.
Another mediaeval text says that the Byzantines are to be considered as penitents among the faithful and that if they choose to join the Orthodox they are not to be baptised.
Thus, it seems to me that the anointing with chrism is very recent, later than 1854.