How do I access the built-in dictionaries on iOS without hitting "Define..."?
This is currently not possible. Apple also forbids third party developers to create a simple wrapper around the dictionary, which means that there will probably not be any applications that do just that. There are, however, several other third party dictionaries which are quite good.

A UIReferenceLibraryViewController object provides a dictionary service to look up the definition of a word or term from within an app. It should not be used to display wordlists, create a standalone dictionary app, or republish the content in any form.

According to this answer Can iOS&#39; builtin dictionary be accessed/browsed as an independent app? you can download the application DictionaryPlusPlus, which does just that, here https://github.com/josh-/DictionaryPlusPlus, it is open source and you will have to compile and sign it yourself.