The only member of your crew with whom you can guarantee attendance to each battle is your first mate. Your first mate is the crew member that you move to the top of your crew roster and will be marked as such when you do. Being first mate equals 100% chance to show up in battle. So, who is the best choice? I currently have a lvl 17 Buccaneer and I have by no means tested every companion in the game (I haven't bought a single one from the crown shop, though I probably will once I reach top level), but at this point, I can't imagine that there is a consistently better companion than Bonnie Anne.
Bonnie Anne is one of the first companions you get, after the one you get for choosing your own class, and she also get the first promotion quest. After that quest, which happens when she turns level 8, you should make her your first mate. You have her set up right, though. I gave her Double Tap, which means she will fire on another enemy after she kills the one she was targeting and Burst Fire (I think that's what it's called), which says that with every successful hit, she has the possibility to take another shot as her epic abilities. With this set-up, I've seen her shoot an enemy, Burst Fire, Burst Fire, mob dies, Double Tap onto another enemy, Burst Fire, Burst Fire, Burst Fire, next mob dies. That was one turn!
I don't know if she is working as intended, but she is may more powerful that any other companion I've seen and, in some ways, more powerful than my Pirate! That's why she's my first mate. If later on I find a replacement, I'll post it here!