Why not use に here? (And if it were used, would the meaning of the sentence change?)
It would sound pretty weird, and arguably be ungrammatical. Your suggested parsing of the sentence,
(その夢の完全なる成就)と(それによる至福の喜び)に, is correct, although I might have an extra set of parenthesis like
((x)と(y)）に. I'm not sure why you find this structure unusual though - it's the exact same thing we'd do in English, like
he basked in X and Y. We don't say
he basked in X in Y, which is kind of what attaching
に to both nouns here would be like. The function of
と here is not really to apply the trailing
に to both as much as it is to join them into one big noun phrase to which the trailing
に can be attached, just like English.
If you really wanted to use
に in both places, something like
(その夢の完全なる成就)にも(それによる至福の喜び)にも is grammatical but still sounds weird to me because it strongly emphasizes that he is basking in both of these things, which in this case doesn't seem to be relevant.
Note that there is a usage of
に that can connect nouns with a meaning like
and (see #8 here), but it's mostly limited to emphasizing that there is something in addition to something else and wouldn't really make sense here.