Child of constraints delay

For a long time now I have been using “child of” constraints in rigging and animation, I had of course noticed, that in some situations the child of constraint has a 1 frame delay, meaning the constrained object seems to lag behind its “parent”
This usually happens when constraining more than one bone of 2 separate armatures, so for example, two characters grappling, usually I “child of” the hands or character 1 to certain bones of the character 2 armature, this seems to work for the most part, except the aforementioned 1 frame lag

I would Use different constraints, but so far , I am happy with “child of” because the “set inverse” and “clear inverse” buttons are VERY useful for me…

So I was wondering, if there is any alternative to the method… Or a way to imitate the “set inverse” and “clear inverse” buttons, for other constraints…

My usual workflow is;
animate some keyframes , add a constraint where needed (click set inverse)key the constraints influence and the bones loc/rot, continue animating with the constraint active, when releasing the constraint, I just key influence to 1 and then 0 on the next frame, (because i use “set inverse” I keep the locations stable)

Effectively I need a magic button to make all constraints work as if they have a “set inverse” button