Its not that the effect "takes precedence", the enhancement is doing what its intended to do:
Charge 1000 gil every double attack proc.
DA proc is determined at the start of the attack round, so even if you smack that wild rabbit for 3000 damage, it still flagged the attack round as having a DA proc, so you still get charged 1000 gil.
I think they went the rather lazy route with it, but at the end of the day, its working exactly the way it says it does.
But even then, it wont do what you were planning to have it do. If your SAM (or nin) has DA+30, you dont jump to Triple attack tier with StoreTP+70. The existing Double Attack (or Triple Attack) you have does not boost Hachimonji's StoreTP->MultiAtk formula in any way.
https://www.bg-wiki.com/ffxi/Hachimonji
Total Store TP Tiers Multiattack per Store TP +1
1~100 Double Attack +1%
101~200 Triple Attack +1%
201+ Quadruple Attack +1%
If you have 30 DA, 15 TA and 130 StoreTP using Hachimonji, I believe these are your final MultiAtk stats:
+130% Double Attack (100 from StoreTP + 30 from DA)
+45% Triple Attack (30 from StoreTP + 15 from TA)
Theres also some negative interaction with HasZanshin I think? I remember a discussion about this, but I dont use the GKT so I didnt care enough to think "this seems important I should remember this".
