In my iCloud e-mail account on my iPad, how can I replace the Trash button with an Archive button?

The archive/trash option is for gmail accounts, you won't get the option for an icloud account : http://support.apple.com/kb/HT4207

Looks like you can archive icloud emails :

- go into Settings > Mail, Contacts, Calendars

- select the icloud account on the right-hand side of that screen

- tap Account (which should have your address next to it) at the top of the next screen

- tap Mail (under 'Advanced') on the popup

- tap Advanced at the bottom of the next popup

- and on the next screen there should be a 'Move Discarded Messages Into' setting

I didn't know that setting was there until you said that you had archive on your phone.