The biggest proof of the Silmarillion being from Bilbo's "Translations from the Elvish" is that there is a note from Tolkien saying that "Note on the Shire Records" should be put in The Silmarillion's preface instead of The Lord of the Rings' prologue.
The Second Edition of the Lord of the Rings, first published in 1965, adds a section to the prologue titled "Note on the Shire Records", which describes how the content in The Lord of the Rings was preserved and transmitted.
In here, Tolkien mentions the "Translations from the Elvish", and says they were about the first age.
But the chief importance of Findegil's copy is that it alone contains the whole of Bilbo's 'Translations from the Elvish'. These three volumes were found to be a work of great skill and learning in which, between 1403 and 1418, he had used all the sources available to him in Rivendell, both living and written. But since they were little used by Frodo, being almost entirely concerned with the Elder Days, no more is said of them here.
The Lord of the Rings - Prologue - "Note on the Shire Records"
At the same time, Tolkien also modified the main text of the book to reference this.
In the evening they went to say good-bye to Bilbo. ‘Well, if you must go, you must,’ he said. ‘I am sorry. I shall miss you. It is nice just to know that you are about the place. But I am getting very sleepy.’ Then he gave Frodo his mithril-coat and Sting, forgetting that he had already done so; and he gave him also three books of lore that he had made at various times, written in his spidery hand, and labelled on their red backs: Translations from the Elvish, by B.B.
The Lord of the Rings - Book VI, Chapter 6 - "Many Partings"
Based on this information, many people including Robert Foster and Christopher Tolkien assumed that this was the in-universe source for The Silmarillion, but Christopher Tolkien did not feel that the evidence was sufficient enough to introduce this concept into the Silmarillion himself.
In The Complete Guide to Middle-earth Robert Foster says: 'Quenta Silmarillion was no doubt one of Bilbo's Translations from the Elvish preserved in the Red Book of Westmarch.' So also I have assumed: the 'books of lore' that Bilbo gave to Frodo provided in the end the solution: they were 'The Silmarillion'. But apart from the evidence cited here [i.e. from The Lord of the Rings itself], there is, so far as I know, no other statement on this matter anywhere in my father's writings; and (wrongly, as I think now) I was reluctant to step into the breach and make definite what I only surmised.
The History of Middle-earth volume I - The Book of Lost Tales Part One - Foreword
However, sometime after this Christopher found considerably better proof.
In the final volume of The History of Middle-earth, Christopher Tolkien notes that his father at some point considered placing the 'Note on the Shire Records' section in The Silmarillion, and not in The Lord of the Rings.
The Note on the Shire Records entered in the Second Edition. In one of his copies of the First Edition my father noted: 'Here should be inserted Note on the Shire Records'; but he wrote against this later: 'I have decided against this. It belongs to Preface to The Silmarillion.' With this compare my remarks in the Foreword to The Book of Lost Tales Part One, pp. 5-6.
The History of Middle-earth volume XII - The Peoples of Middle-earth - "The Prologue"
In the event, Tolkien obviously changed his mind about part of this, as the 'Note on the Shire Records', was added into the second edition of The Lord of the Rings, but this remains as proof that there was a point relatively late in Tolkien's life (probably around the 1960s) where he felt The Silmarillion was transmitted by The Hobbits the same way The Lord of the Rings was transmitted.