NO unit may battle more than once in a single battle (any Cav breakthrough and second battle is a "new" battle).
This is a iron/non changeable basic rule, which apply not only to the FS card, but also to ALL other existing or future rules (not only command cards and tactician cards, but also to any other special rules, which have the same effect) and therefore the FAQ is found in the Basic & Battle
www.commandsandcolors.net/napoleonics/th.../faqs/57-battle.html
First entry
So only the order of battle change, defender first instead second and versa
Yeah the "battle first" should give it away but better phrasing would help.
It´s not always the card phrasing. The familiar C&C players (who plays also some ohter first CC games e.g. MM44) know this, because in each system
Battlecry, Memoir 44, CCA, Battlelore, we have/had always the same "special players" interpretations, and in all these CC games we have already the same FAQ answer
And i can bet with you, if we got the next CC system (AWI), and there is also a First Strike Card, we will get the same "special player" interpretation again because they say any like this "You can´t compare this new FS not with the FS cards of the existing other C&C systems, because we have now a complete new game).
PS. and here one of the threads (there are some more) on BGG, which answered it already (and said in short form, what i say above with some more words

)
www.boardgamegeek.com/article/6270359#6270359