They don't mean exactly the same thing. Anyway stick with best. Bosom as a modifier of friend is extremely old-fashioned. My mother used it occasionally but I haven't heard it in at least fifty years.
At least in the US, the term is 'bosom buddy' not 'bosom friend'.
That set phrase is still used some, probably more so than the term 'bosom' alone is used to a woman's chest nowadays. That usage is quite outdated*. I've never heard of it used as a modifier outside of that set phrase 'bosom buddy'.
*Although Greg Brown does sing about nearly getting "bosumed to death" by his aunts":lol:
To me, "bosom buddies" sounds like an old-fashioned way to say "best (male) friends". I don't think anyone has used it in colloquial American English for decades, except ironically.