[The Saucepan Man]

Could not have Arwen gone with him, instead of just fading away in Lorien?

Yes, I don't see why not. But, having chosen mortality, she would remain mortal and so would eventually die in Valinor, just as Bilbo, Frodo, Sam and (presumably) Gimli did. I doubt that she would have been too keen to put her father and other relatives through the pain of separation a second time.