In some themes after a cart ajax call runs, then just target the cart menu button and .click() it with javascript to open the related cart drawer that’s wired to such a button.
Some themes have a dedicated function|event to open drawers.
Note how this in no way guarantees the cart-drawer has up to date information of the ajax calls that were just made.
Again some themes may have functions/events to internally update the cart-drawer contents in addition to opening.
If not that is an advanced theme customization to change/copy the themes underlying cart behaviors to take programatic control of the contents update.