"Bro" is traditionally the male equivalent of "sis" even if more recently it has come to have other uses. "Brother"/"sister" sounds horribly stiff if misused (which it often is in these patches) but is appropriate in many circumstances.
In most cases, a person should use "my brother"/"my sister" when referring to a sibling in 3rd person (e.g. "I just saw my brother over there.") or in an internal dialogue (e.g. "I'm hugging my brother.") and use the sibling's given name when talking directly to them (e.g. "Carl, hug me.") I think that's the best way to handle it.
Edit: Also, to give a little informality but avoid the more modern interpretation of "bro", you can use "little bro"/"big bro"/"little sis"/"big sis" when appropriate. These can be used when talking directly between siblings, but shouldn't be overused.