Shri Rama knew that Dasaratha promised to the father of Kaikeeyi at the time of her marriage to Dasaratha, that Dasaratha would confer his (Dasaratha's) kingdom as dowry to the father of the girl.
2-107-3 Sanskrit Verse
Puraa bhraataha pitaa naha
sa maataram tee samudvahan
raajya shulkam anuttamam
Bharata approached Rama at Chitrakoot with a request to return to Ayodhya. One of the replies given by Rama to Bharata was that their father Dasaratha promised to Bharata's maternal grand father that he (Dasaratha) was giving his kingdom as dowry to her father. Rama wanted their father's promise to be kept.
Brother! In the past, while marrying your mother, our father promised your maternal grand father that he would confer an exceptional "rajya s`ulkam" (kingdom as dowry).
When Dasaratha gave away his kingdom as dowry to the Kaikeeyi's father: Asvapati, Dasaratha had no right on the kingdom. He was only a representative/fief of Asvapati. Dasaratha knew that he was breaking his promise to Kaikeeyi's father, by proposing the coronation of Rama S/o Kausalya.
At 2-2-16 Dasaratha expressed his desire to coronate Rama, in his assembly. There he did not disclose the promise made by him to Asvapati and ask the opinion of the Assembly whether it will be correct to break a promise. Kaikeeyi's father was a powerful king with a strong cavalry. Dasaratha should have discussed the "military" consequences if As`vapati (Kaikeyi's father) retaliated, irritated by the broken promise.
At 2-3-8, sage Vasisht`a ought to have raised the issue of Dasaratha's promise to Kaikeeyi's father. But Vasisht`a goes on with the preparations for the coronation. Vasisht`a is a great ancient sage who is believed to have super-human powers. Hence, there cannot be a question of his not knowing about Dasaratha's promise.
At 2-3-48, Rama hearing Dasaratha's proposal to coronate him (Rama), just folds his hands to his father. Rama ought to have raised the issue of promise made to Kaikeeyi's father. If he was unable to tell his father directly, he could have approached the Guru Vasisht`a and approached his father for clarification. It means that Rama was willing to be coronated before Bharata came and his grandfather raised objections.
At 2-4-25, Dasaratha wants the coronation to take place while Bharata is away from the city. This shows that Dasaratha is apprehensive that Asvapati or Bharata will object.
At 2-4-27, Dasaratha expresses a suspicion that Bharata may change his mind, howsoever righteous he may be.
At 2-4-28, Rama should have questioned his father "Why should Bharata change his mind?". But Rama does not question because he is aware that Bharata has a legitimate claim on the kingdom and that he may raise it. Hence Rama salutes his father (probably for bestowing a secret benefit).
If Dasaratha had given his kingdom as dowry to Asvapati, Rama would not have had a right to occupy the throne even after returning from 14 years exile. After returning from exile, Rama joyfully took back his throne from Bharata and ruled it for thousands of years. Then what would have happened to the promise made by Dasaratha to Asvapati, giving his kingdom as dowry in exchange for Kaikeeyi?