All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The narrated Thikr phrases (mention of Allaah) to be recited after the prayer are "Subhanah Allah (Glory be to Allah), Al-Hamdu Lillah (praise be to Allah), and Allahu Akbar (Allah is the Greatest). As far as we know, the additional phrases of ‘wa bihamdihi’ along with the Tasbeeh (Subhanah Allah), and Rabil-‘Aalameen (Lord of the Worlds)" when saying the Tahmeed (Al-Hamdu Lilllah) do not exist.
Therefore, it is sufficient for you to say what is confirmed in the Sharee’ah without adding anything to it.
Indeed, the Prophet objected to Al-Baraa’ Ibn 'Aazib
when he added to a specific Thikr that which is not part of it; as in the Hadeeth when he
was teaching him what to say when going to sleep: “Al-Baraa’
said: “As I repeated these words in order to memorize them, I said: “I affirm my faith in Your Messenger whom You have sent.” He
said: Say: “I affirm my faith in the Prophet whom You have sent. [i.e., Prophet not Messenger]” [Al-Bukhari and Muslim]
Moreover, it was reported that Ibn Mas’ood objected to the one who added the wording: “Wahdahu la shareeka lahu”, in the Tashahhud while he was teaching him: At-Tabarani reported that Al-'Ala’ ibn Al-Musayyib narrated that his father said: “'Abdullah ibn Mas'ood was teaching a man the Tashahhud and he said: “Ashhadu alla ilaha illa Allah, wa ashhadu anna Muhammadan 'abduhu wa rasooluhu” (I testify that None has the right to be worshiped but Allah, and I testify that Muhammad is His slave and His Messenger); and the man said (after Ashhadu alla ilaha illa Allah): “…wahdahu la shareeka lahu (the One Who has no partners); thereupon 'Abdullah ibn Mas'ood said to him: “Indeed, He [Allah] is as such (without partners), but we should stick to that which we have been taught.”
As regards adding Al-‘Aliyyul-‘Atheem when saying La hawla wala quwwata illa billah (there is no might nor power but with Allah) when repeating the words of the Athan, then this was reported in some narrations.
In the Musannaf of 'Abur-Razzaq on the authority of ‘Abdullah ibn Al-Harith ibn Nawfal, the Prophet used to say when he heard the Mu'aththin (the person who calls the Athan) saying “Allahu Akbar (Allah is the Greatest)”, he
would say: “Allahu Akbar”; when he said: “Ashhadu alla ilaha illa Allah” (I testify that None has the right to be worshiped but Allah); he
would say: “Ashhadu alla ilaha illa Allah”; and when he said: ashhadu anna Muhammadan rasoolu Allah” (I testify that Muhammad is the Messenger of Allah); he
would say the same thing, and when he [the Mu'aththin] says: “Hayya ‘ala as-salah (come to the prayer) he
would say: “la hawla wala quwwata illa billah Al-‘Alyyi Al-‘Atheem (there is no might nor power except with Allah, the Most High, the Greatest.” [End of quote]
Ibn Qudaamah said in his book Al-Muqni’: “Except when the Mu'aththin says: Hayya ‘ala as-salaah and hayya ‘ala al-falah (Come to prayer; come to success); one should say: la hawla wala quwwata illa billahi Al’-Alyyi Al’-Atheem.” [End of quote]
When explaining the book Al-Muqni’, Ibraheem ibn Al-Muflih, said in his book Al-Mubdi’ sharh Al-Muqni':
“In regard to (Al’-Alyyi Al’-Atheem’; I investigated the matter and I found it in Al-Musnad (authored by Imam Ahmad) from the Hadeeth of Abu Raafi’ that when the Prophet heard the Mu'aththin, he would said the same thing, until he [Mu'aththin] says: 'Hayya 'ala as-salaah (come to the prayer), Hayya 'ala al-falaah’ (come to success),' he
would say: 'La hawla wa la quwwata illa billah Al-‘Alyyi Al-‘Atheem (There is no power nor might except with Allah the Most High, the Greatest);' [At-Tabarani in his book Al-Mu’jam Al-Kabeer, but its chain of narrators is not sound (i.e., an inauthentic narration)]” [End of quote]
To conclude, this addition was reported in some narrations, contrarily to the addition ‘wa bihamdihi’ and ‘rabi Al-‘Aalameen’, in the Thikr after the prayer; this additional phrase was not reported in any narration, so it is not permissible to recite it.
Allah knows best.