Two quotations from Srimad-Bhagavatam are relevant here:
prAyena munayo rajan nivrittA bidhisedhtah (2//7) meaning that Most great saints are above the scriptural do-s and don't-s.
dharma-vyatikramo drishta isvaraNam cha sAhasam/ tejiyasAm na doshAya vahNeh sarvabhujo yathA// (10/33/29) meaning : Exception to the laws of righoutness and courage are seen in the Most Powerful Ones. For the extremely Vigorous individuals, these are not to be blamed as the fire eats everything but remains pure.
So great souls liie Swamiji are above the scriptural laws and so never need prayaschitta.
By the way,Swamiji had to take meat for three reasons:
He lived on 'viksha' and had to eat whatever is offered as food in the West. He had little personal choice.A monk does not have to be bothered about food according to the Sannyasa Upanishad (see below) as mentioned about Sri Chaitanyadeva. Mahaprabhu Nityananda also ate fish and meat as per Sri Chaitanya Bhagavata.
A Paramahansa sannyasin has to rise above every sense of duality.
He suffered from severe diabetes which had no treatment in those days.He was advised by the doctors to live primarily on meat.
On 10 November 1896 He spoke in London in His lecture on 'Practical Vedanta':
Some people have been kind enough to start an antivivisection society. I asked a member, "Why do you think, my friend, that it is quite lawful to kill animals for food, and not to kill one or two for scientific experiments?" He replied, "Vivisection is most horrible, but animals have been given to us for food." Oneness includes all animals. If man's life is immortal, so also is the animal's. The difference is only in degree and not in kind. The amoeba and I are the same, the difference is only in degree; and from the standpoint of the highest life, all these differences vanish. A man may see a great deal of difference between grass and a little tree, but if you mount very high, the grass and the biggest tree will appear much the same. So, from the standpoint of the highest ideal, the lowest animal and the highest man are the same. If you believe there is a God, the animals and the highest creatures must be the same. A God who is partial to his children called men, and cruel to his children called brute beasts, is worse than a demon. I would rather die a hundred times than worship such a God. My whole life would be a fight with such a God But there is no difference, and those who say there is, are irresponsible, heartless people who do not know. Here is a case of the word practical used in a wrong sense. I myself may not be a very strict vegetarian, but I understand the ideal. When I eat meat I know it is wrong. Even if I am bound to eat it under certain circumstances, I know it is cruel. I must not drag my ideal down to the actual and apologise for my weak conduct in this way. The ideal is not to eat flesh, not to injure any being, for all animals are my brothers. If you can think of them as your brothers, you have made a little headway towards the brotherhood of all souls, not to speak of the brotherhood of man! That is child's play. You generally find that this is not very acceptable to many, because it teaches them to give up the actual, and go higher up to the ideal. But if you bring out a theory which is reconciled with their present conduct, they regard it as entirely practical.
In another place He says:
There they used to make much fuss about my food and say, "Why should you eat that food or eat from the hands of such and such?" — and so on. To which I had to reply, "I am a Sannyasin and a mendicant friar and what need have I to observe so much outward formality with regard to food etc.? Do not your scriptures say, " One should beg one's food from door to door, ay even from the house of an outcast"? But of course external forms are necessary in the beginning, for the inner realisation of religion, in order to make the truth of the scriptures practical in one's life. Haven't you heard of Shri Ramakrishna's story of "wringing out the almanac for water"? Outward forms and observances are only for the manifestation of the great inner powers of man. The object of all scriptures is to awaken those inner powers and make him understand and realise his real nature. The means are of the nature of ordinances and prohibitions. If you lose sight of the ideal fight over the means only, what will it avail? In every country I have visited, I find this fighting over the means going on, and people have no eye on the ideal. Shri Ramakrishna came to show the truth of this.
Realisation of the truth is the essential thing. Whether you bathe in the Ganga for a thousand years or live on vegetable food for a like period, unless it helps towards the manifestation of the Self, know that it is all of no use. If on the other hand, any one can realise the Atman, without the observance of outward forms, then that very non-observance of forms is the best means. But even after the realisation of Atman, one should observe outward forms to a certain extent for setting an example to the people. The thing is you must make the mind steadfast on something. If it is steadfast on one object, it attains to concentration, that is, its other modifications die out and there is a uniform flow in one direction. Many become wholly preoccupied with the outward forms and observances merely and fail to direct their mind to thoughts of the Atman! If you remain day and night within the narrow groove of ordinances and prohibitions, how will there be any expression of the soul? The more one has advanced in the realisation of the Atman, the less is he dependent on the observances of forms. Shankaracharya also has said, " Where is there any ordinance or prohibition for him whose mind is always above the play of the Gunas?" Therefore the essential truth is realisation. Know that to be the goal. Each distinct creed is but a way to the Truth. The test of progress is the amount of renunciation that one has attained. Where you find the attraction for lust and wealth considerably diminished, to whatever creed he may belong, know that his inner spirit is awakening. The door of Self-realisation has surely opened for him. On the contrary if you observe a thousand outward rules and quote a thousand scriptural texts, still, if it has not brought the spirit of renunciation in you, know that your life is in vain. Be earnest over this realisation and set your heart on it. Well, you have read enough of scriptures. But tell me, of what avail has it been? Some perhaps thinking of money have become millionaires, whereas you have become a Pundit by thinking of scriptures. But both are bondages. Attain the supreme knowledge and go beyond Vidyâ and Avidyâ, relative knowledge and ignorance. (Svami-Sishya-samvad, CW).
The meaning of the word Ishvara for the Bhagavata-sloka could be seen from : https://www.sanskritdictionary.com/?iencoding=iast&q=ईश्वर&lang=sans&action=Search
As per the answer What is the full form of this sloka?, the Sannyasa Upanishad says:
Air is not spoiled by touching (any object); fire by the activity of burning; waters, by urine and faeces (getting into them); and a mendicant monk by faults ('Dosha') in food.
Sri Chaitanyadeva also mentions that an GREAT AND ABLE (MahA-AdhikAri) person is beyond all types of restrictions:
suno vipra! yadi mahA-adhikAri hay/tabe tAn guna doshkichu nA janmay//(Sri Sri Chaitanya-Bhagavata, Shesha-Khanda, chapter 7).
Mahaprabhu quotes a sloka(20/36) from Srimad-Bhagavatam from Canto 11 to support His starement that says that real saints are above all virtues and vices.
This also proves that for Swamiji, no prayaschitta was needed at all.