A friend needs to know this in British English, so I'd be most interested in replies from Brits. (Or please say so if you're from somewhere else.)
THANK YOU ALL. So, it's either hyphenated or even better: "register again".
And just for the record, I've looked up what Fowler (1926) says: In re(-) compounds, the hyphen is usual before e (re-entrant, re-examine, &c.); not uncommon before other vowel (re-armament or rearmament, reiterate or re-iterate, reorganize or re-organize, re-urge or reurge), especially when the look of the word, as in the first & fourth examples, is deceptive or puzzling without it; common when the compound is used after the simple word (make & re-make, discussion & re-discussion); & necessary when a modern compound such as re-cover = put a new cover upon, re-pair = pair afresh, or re-count = count again, is to be distinguished from a better known & differently pronounced old word (recover get back, repair mend, recount narrate).
And the Oxford Dictionary of English (2005): In modern English, the tendency is for words formed with prefixes such as re- to be unhyphenated: restore, remain, reacquaint. One general exception to this is when the word to which re- attaches begins with e: in this case the hyphen is often inserted for clarity: re-exanime, re-enter, re-enact. A hyphen is sometimes also used where the word formed with the prefix would be identical to an already existing word: re-cover (meaning 'cover again', as in we decided to re-cover the dining-room chairs) not recover (meaning 'get better in health').
So both fail to address the question whether the word is already a "proper" word, or whether the first three or four letters form other well-known word, but I think you're right in taking that in account as well.