Since「するように」and「あるように」are valid, I guess you are talking about the -です form.
The word before 〜ように should be in 連体形.
However, です's 連体形 are only used before ので or のに.
I think there are no logical explanation for this.
にて is simply a formal way of saying で (as in the particle for location or means). It's common in announcements and official documents. I don't know its linguistic history, but if you search the web you can occasionally see the name of this song translated as "In Zanarkand" or "At Zanarkand", which is more literally. I think "To Zanarkand" just sounds more ...