I was referring to currently acceptable spellings, as you know the English language is constantly evolving and both the Cambridge English dictionary and the freedictionary use hairbrained. Initially either spelling was normal.

Yes, but that was at a time when spelling was very fluid - people largely wrote what they heard.  So, some people would spell the word "hare", others "hair".  However, by Johnson's time (and largely thanks to the introduction of the printing press), the spelling had become standardised as "hare".  To spell the word "hairbrained" is to lose the core meaning of the word.