For an actual sibling, you'd nearly always use deartháir. Note that it becomes dheartháir only in the right context, as with the expression mo dheartháir ("my brother"), where the initial sound/spelling change occurs because the word "mo" is preceding it.

Bríd is a native speaker, so if she feels that you could use deartháir for a very close non-sibling, that's good enough for me. In a sense, you'd be emphasizing that he's a "real brother" (even if not actually related), given the etymology of the word as explained above.

As already explained, bráthair is most commonly used now in Ireland for religious brothers, but it is also used in expressions like "brothers in arms" and in some older proverbs and sayings (and in the Bible) for siblings. It is still used in Scottish Gaelic for sibling brothers, just as it once was long ago in Ireland (with the accent reversed in Scottish Gaelic: bràthair).

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.