Note that Jesus was not specifically talking about the official but about the Jews in general in John 4:48
Unless you [people] see signs and wonders, you [people simply] will not believe.
To balance this, we also have John 10:38
But if I do them, even though you do not believe me, believe the works, that you may know and understand that the Father is in me, and I in the Father."
Paul expressed a similar sentiment in 1 Corinthians 1:22
Jews demand signs and Greeks look for wisdom,
Jesus point can be found in John 20:29
Then Jesus told him [Thomas], "Because you have seen me, you have believed; blessed are those who have not seen and yet have believed."
Was Jesus expecting the official to believe in Him without the healing of his son?
I don't think so, at least not as a matter of expectation.
When the official heard John 4:
50a “Go,” Jesus replied, “your son will live.”
He believed before seeing the result:
50b The man took Jesus at his word and departed.
And his household was blessed:
53Then the father realized that this was the exact time at which Jesus had said to him, “Your son will live.” So he and his whole household believed.