If Rāma could sacrifice his kingdom to Bharata, why couldn't Yudhiṣṭhira do it for
Duryodhana?
Because Bharata was righteous while Duryodhana was unrighteous.
It can be observed from many events of life of these two.
Bharata
Bharata was born from the portion of Lord Vishnu who always incarnates to establish Dharma. It is mentioned in Ramayana, Bala Kanda, Sarga 18.
भरतो नाम कैकेय्याम् जज्ञे सत्य पराक्रमः |
साक्षात् विष्णोः चतुर्थ भागः सर्वैः समुदितो गुणैः || १-१८-१३
Queen Kaikeyi gave birth to Bharata, one embodied with all merits, and whose truthfulness itself is his valour and who is fourth component of manifest Vishnu, namely Rama. [1-18-13]
And also Lord Rama has described some characteristics of Bharata in the same chapter of Ayodhya Kanda which you have mentioned in your question, i.e., Valmiki Ramayana, Ayodhya Kanda, Sarga 97.
मन्येऽहमागतोऽयेध्यां भरतो भ्रातृवत्सलः |
मम प्राणात्र्पियतरः कुलधर्ममनुस्मरन् || २-९७-९
श्रुत्वा प्रव्राजितं मां हि जटावल्कलधारिणम् |
जानक्यासहितं वीर त्वया च पुरुषर्षभ || २-९७-१०
स्नेहेनाक्रान्तहृदयः शिकेनाकुलितेन्द्रियः |
द्रष्टुमभ्यागतो ह्येष भरतो नान्यथऽऽगतः || २-९७-११
"I think Bharata was back to Ayodhya, with full of affection for his brothers. He is dearer to me than my life- he who is mindful of the duties of his race. Hearing of my exile and that I was wearing matted locks and the antelope skin, accompanied by Seetha and you, O most valiant of warriors, in his devotion towards me and due to the distress that troubles his mind, Bharata has come to see me. He has not come with any other motive."
अम्बाअं च कैकयीं रुष्य परुषं चाप्रियं वदन् |
प्रसाद्य पितरं स्रीमान् राज्यं मे दातुमागतः || २-९७-१२
"Getting angry with Kaikeyi his mother and speaking unkind and harsh words to her, the glorious Bharata after getting permission from our further, came here to bestow kingdom to me."
विप्रियम् कृत पूर्वम् ते भरतेन कदा न किम् |
ईदृशम् वा भयम् ते अद्य भरतम् यो अत्र शन्कसे || २-९७-१४
"Has Bharata at any time done any harm to you previously? What so the reason for you to be so apprehensive of Bharata now?"
उच्यमानो हि भरतो मया लक्ष्मण तत्त्वतः |
राज्यम् अस्मै प्रयग्च्छ इति बाढम् इत्य् एव वक्ष्यति || २-९७-१८
"O, Lakshmana! When Bharata receives this command from me, "Cede the empire to him", he will answer, So, be it."
There are still more verses to support his righteousness.
Duryodhana
According to Mahabharata, Adi Parva, SECTION LXVII, Duryodhana was the incarnation of Kali demon. Hence he had some inherent properties of Kali.
The evil-minded and wicked king Duryodhana, the destroyer of the fair fame of the Kurus, was born of a portion of Kali on earth. He it was who caused all creatures to be slain and the earth to be wasted; and he it was who fanned the flame of hostility that ultimately consumed all.
Duryodhana was jealous of Pandava-s, especially Bhima. He even tried to poison Bhima in food, as described in Mahabharata, Adi Parva, SECTION CXXVIII.
Meanwhile the wicked Duryodhana had mixed a powerful poison with a quantity of food, with the object of making away with Bhima. That wicked youth who had nectar in his tongue and a razor in his heart, rose at length, and in a friendly way fed Bhima largely with that poisoned food, and thinking himself lucky in having compassed his end, was exceedingly glad at heart.
Duryodhana attempted to kill all the Pandavas. He had almost burned them in Lakshagriha as described in Mahabharata, Adi Parva, SECTION CXLVI.
The Pandavas have, by Dhritarashtra, been sent to Varanavata, where they will, at Dhritarashtra's command, enjoy themselves during the festivities. Do that by which thou mayest this very day reach Varanavata in a car drawn by swift mules. Repairing thither, cause thou to be erected a quadrangular palace in the neighbourhood of the arsenal, rich in the materials and furniture, and guard thou the mansion well (with prying eyes). And use thou (in erecting that house) hemp and resin and all other inflammable materials that are procurable. And mixing a little earth with clarified butter and oil and fat and a large quantity of lac, make thou a plaster for lining the walls, and scatter thou all around that house hemp and oil and clarified butter and lac and wood in such a way that the Pandavas, or any others, may not, even with scrutiny behold them there or conclude the house to be an inflammable one. And having erected such mansion, cause thou the Pandavas, after worshipping them with great reverence, to dwell in it with Kunti and all their friends. And place thou there seats and conveyances and beds, all of the best workmanship, for the Pandavas, so that Dhritarashtra may have no reason to complain.
Last but not least, Yudhishthira had asked for only five villages for the livelihood instead of whole kingdom, but that request was also declined by Duryodhana.
Thou hast also heard all that was said unto Sanjaya. O thou of great effulgence, these words were even these, 'Let only five villages be granted to us, viz., Avishthala, and Vrikasthala, and Makandi, and Varanavata, and for the fifth, any other,--O thou of mighty arms, O Kesava, even this was the message that was to have been delivered to Duryodhana and his counsellors. But, O Krishna, O thou of Dasarha's race, hearing those words of Yudhishthira, endued with modesty and anxious for peace, Suyodhana hath not acted according to them.
Duryodhana disrespected his elders like Bhishma, Drona, Vidura. He also ill-treated lord Krishna when he was a peace keeper on behalf of Pandavas.
So, Lord Rama knew that Bharata can be a good king because of his good qualities. Similarly, Yudhishthira also knew that Duryodhana cannot be a good king due to his wicked qualities.
