I'm not sure how to handle these BST JAs, but I tried to test their Enmity. All of BST's JAs are pretty ridiculously low-Enmity. It appears to me that Fight, Familiar, Ready, and Spur all give 0 CE/VE.

Next I built up 1000 CE (8x Threnody on BST/BRD, 10x Threnody on WHM mule) and Snarl'ed away my hate on the BST.
10 Threnodies shifted hate back to the BST, and then 10x Cures for 0 HP shifted it back to the WHM (1010 CE). So Snarl shifted 99% of my Enmity to the pet.

I have been working through the stuff on BGwiki to find out what we don't actually know yet and to correct things that showed up on ffixclopedia without citation/testing.