In the Analysis Window Browse tab, the drop-down options menu includes at the bottom Synch Browser on Verse Change. I think that's the setting; but you may already have that.
I'm puzzled about a Toolbar button to sync these two. In fact, as I poke into this, I'm puzzled about a lot of things. In Help section 23, Analysis Window: The Browse Tab, under The Browse Tab Header, it says, "To change the items displayed right click on a blank area of the header and select the options that you want from the context menu," and then gives a list of options. I can't for the life of me find that context menu by right-clicking in the header of the Analysis Window Browse tab. All I get is a 3-item menu offering Outline Heading, Browse List Boxes, and Popup CL. Also, at the end of sec. 23, it says, "If you scroll the Browse Tab window [with sync turned off] and then want to synchronize the Browse Tab to the Browse Window once again, you can press the Synchronize Tab Browser button on the Browse Tab header." That sounds like the button you mentioned, but I can't find it either.
So, RandomSF, it seems that I need more help than you do! But in fact I don't think there is a keyboard shortcut to do this. Those are listed in Help section 72.