al-Salamu `Aleykum,
Why `Ali mentioned the prohibition of the meat of domestic donkey on Khaybar to ibn `Abbas in the popular narration?
The reason is that ibn `Abbas did not view it as prohibited:
حَدَّثَنَا إِسْمَاعِيلُ بْنُ يَحْيَى الْمُزَنِيُّ، قَالَ: ثنا مُحَمَّدُ بْنُ إِدْرِيسَ، قَالَ: ثنا سُفْيَانُ، قَالَ: أَخْبَرَنَا عَمْرٌو، قَالَ: قُلْتُ لِجَابِرِ بْنِ زَيْدٍ: " إِنَّهُمْ يَزْعُمُونَ أَنَّ النَّبِيَّ قَدْ نَهَى عَنْ لُحُومِ الْحُمُرِ الأَهْلِيَّةِ " فَقَالَ: قَدْ كَانَ يَقُولُ ذَلِكَ، الْحَكَمُ بْنُ عَمْرٍو الْغِفَارِيُّ، عَنِ النَّبِيِّ " أَبَى ذَلِكَ الْحَبْرُ، يَعْنِي ابْنَ عَبَّاسٍ رَضِيَ اللَّهُ عَنْهُمَا وَقَرَأَ: قُلْ لا أَجِدُ فِي مَا أُوحِيَ إِلَيَّ مُحَرَّمًا عَلَى طَاعِمٍ يَطْعَمُهُ
[Isma`eel bin Yahya al-Muzani told us, Muhammad bin Idrees told us, Sufiyan told us, `Amro told us, he said: I said to Jabir bin Zayd (the companion of ibn `Abbas): "They claim that the Prophet (saw) prohibited the meat of domestic donkeys." He replied: "Al-Hakam bin `Amro al-Ghifari (a Sahabi) used to attribute this to the Prophet (saw), but that scholar -meaning ibn `Abbas- refused and recited: {Say, I do not find within that which was revealed to me [anything] forbidden to one who would eat}.]
Meaning, he refused that the meat of the donkey is completely forbidden although he never denied the narrations, he just explained them differently as you will see below.
حَدَّثَنَا ابْنُ أَبِي دَاوُدَ، قَالَ: ثنا عَبَّادُ بْنُ مُوسَى الْخُتَّلِيُّ، قَالَ: ثنا يَحْيَى بْنُ سَعِيدٍ الأُمَوِيُّ، عَنِ الأَعْمَشِ، قَالَ: حُدِّثْتُ عَنْ عَبْدِ الرَّحْمَنِ بْنِ أَبِي لَيْلَى، قَالَ: قَالَ ابْنُ عَبَّاسٍ رَضِيَ اللَّهُ عَنْهُمَا: مَا نَهَى رَسُولُ اللَّهِ يَوْمَ خَيْبَرَ عَنْ أَكْلِ لُحُومِ الْحُمُرِ الأَهْلِيَّةِ إِلا مِنْ أَجْلِ أَنَّهَا ظَهْرٌ
[Ibn abi Dawoud told us, `Abbad bin Musa told us, Yahya bin Sa`eed told us, from al-A`mash, I was told that `Abdul-Rahman bin abi Layla said: Ibn `Abbas said: "The Prophet (saw) never forbade the meat of domestic donkeys on Khaybar except because they needed them to carry things."]
Ibn `Abbas's opinion then seems to have shifted to "Tawaqquf", meaning he would refrain from saying anything with regards to it being Halal or Haram.
كَمَا رُوِّينَا مِنْ طَرِيقِ الْبُخَارِيِّ: نَا مُحَمَّدُ بْنُ أَبِي الْحُسَيْنِ، نَا عُمَرُ بْنُ حَفْصِ بْنِ غِيَاثٍ، نَا أُبَيٌّ، نَا عَاصِمُ بْنُ أَبِي النَّجُودِ، عَنْ عَامِرِ الشَّعْبِيِّ، عَنِ ابْنِ عَبَّاسٍ، أَنَّهُ قَالَ: " لَا أَدْرِي أَنَهَى عَنْهُ رَسُولُ اللَّهِ مِنْ أَجْلِ أَنَّهُ كَانَ حَمُولَةَ النَّاسِ فَكَرِهَ أَنْ تَذْهَبَ حَمُولَتُهُمْ، أَوْ حَرَّمَهُ فِي يَوْمِ خَيْبَرَ لَحْمَ الْحُمُرِ الْأَهْلِيَّةِ
[From the path of Bukhari, Muhammad bin abi al-Hasan told us, `Umar bin Hafs bin Ghayath told us, my father told me, `Asim bin abi al-Najoud told us, from `Amir al-Sha`bi, from ibn `Abbas that he said: "I do not know if the Prophet (saw) forbade it because they can carry the people's things (during Khaybar), or they were forbidden on the day of Khaybar because they were domestic donkeys."]
Because ibn `Abbas had these odd Fatwas, `Ali spoke to him saying: "You are a lost man, did you not hear that the Prophet (saw) forbade the Nikah of Mut`ah and the meat of the domestic donkey on the day of Khaybar."