Unfortunately there is no out-of-the-box feature that will allow you to do exactly this.
There is an easy way to attach a handler for the Tapped event of the overflow button (via the OverflowButtonTemplate). However, even if you were to attach a Tapped handler and opened the side drawer, you cannot prevent the popup from opening.
One way to go here is to overlay a custom button on top of the TabView. You can place this custom button in the top right corner of the TabView. This way the user will not be able to see or click the TabView's overflow button, but only the custom one. In its Tapped handler you can open the side drawer. I am attaching a demo code I tested with.
Note that the work-around is essentially to set the same size to the TabView's overflow button and the custom button, so that the custom button does not overlap the tab items, and so that the user cannot interact with the TabView's overflow button.
There is one thing worth mentioning here - in case all tab items fit in the given area, the TabView's overflow button will not be displayed, but the custom one will be, leading to it overlapping the tab items. If at times you expect this to happen, i.e. the TabView not to have many tab items and the TabView's overflow button not to be displayed all the time, I could investigate if there is a way to detect when the button gets collapsed.
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