This question has been well discussed. See for example Kenner Chapter 13. Here is my take. All the quotes are from the penultimate chapter.
That is the ending Bloom desires. Bloom gets the son he lost.
Bloom desires Stephen to move in and rent Rudy's room. In Bloom's mind, Stephen can get singing lessons from Molly, tutor Molly in Italian, etc. Bloom dreams of long conversations with Stephen, perhaps even a concert featuring Bloom and Stephen.
What various advantages would or might have resulted from a
prolongation of such an extemporisation?
For the guest: security of domicile and seclusion of study. For the
host: rejuvenation of intelligence, vicarious satisfaction. For the
hostess: disintegration of obsession, acquisition of correct Italian
Bloom would like it to be Bloom and Stephen, with Stephen in the role as son. But that is a power relation, even a capture. Consider the various meanings of asylum.
Was the proposal of asylum accepted?
Promptly, inexplicably, with amicability, gratefully it was declined.
While Stephen was not initially explicitly opposed to the idea, upon entering the premises with Bloom, he begins to experience Bloom's dream in the second to last chapter.
This is not the relationship that Stephen is seeking. Stephen certainly has a gaping father-hole and senses the missing father in Bloom.
He heard in a profound ancient male unfamiliar melody the accumulation
of the past.
But Stephen does not need another father. That is a relationship he must move beyond as he becomes a man.
And his subconscious gives this away. As noted in the OP, Stephen sings a horrible anti-Semitic song about a Christian boy who is captured by a Jew and eventually killed! Given that Bloom is a Jew, this is a shot fired.
Then out there came the jew’s daughter
And she all dressed in green.
“Come back, come back, you pretty little boy,
And play your ball again.”
“I can’t come back and I won’t come back
Without my schoolfellows all.
For if my master he did hear
He’d make it a sorry ball.”
She took him by the lilywhite hand
And led him along the hall
Until she led him to a room
Where none could hear him call.
She took a penknife out of her pocket
And cut off his little head.
And now he’ll play his ball no more
For he lies among the dead.
Stephen becomes aware that such a relationship is dangerous for him. What he will get from Bloom comes at a mortal cost.
Condense Stephen’s commentary.
One of all, the least of all, is the victim predestined. Once by inadvertence twice by design he challenges his destiny. It comes when he is abandoned and challenges him reluctant and, as an apparition of hope and youth, holds him unresisting. It leads him to a strange habitation, to a secret infidel apartment, and there, implacable, immolates him, consenting.
If Stephen accepts Bloom's offer, he will again be the predestined victim. The offer comes when Stephen is abandoned, and the apparition of being able to remain a boy is tempting. But if Stephen yields and moves in, he will ultimately be sacrificed.
Stephen cannot accept Bloom's offer. While it is tempting to stay a boy, to stay a boy will "kill" him. Rather he must return and face the world. He must move forward, from the known boy to the unknown man.
Did Stephen participate in his dejection?
He affirmed his significance as a conscious rational animal proceeding
syllogistically from the known to the unknown and a conscious rational
reagent between a micro and a macrocosm ineluctably constructed upon
the incertitude of the void.