I don't think Chatgpt can solve this, it will try to solve things from AvalonDock side of things and not how Revit's using it.
For getting to the Tab's views I use (simplified):
Visual tryVisual = HwndSource.FromHwnd(this._iUICApp.MainWindowHandle).RootVisual;
if (tryVisual is MainWindow)
{
MainWindow uiRvtWindowRoot = (MainWindow)tryVisual;
DockingManager tabDockManager = MainWindow.FindFirstChild<DockingManager>(uiRvtWindowRoot);
LayoutDocumentPaneGroupControl paneTabGroup = MainWindow.FindFirstChild<LayoutDocumentPaneGroupControl>(uiRvtWindowRoot);
Dim tabItems = PaneTabGroup.FindVisualChildren(Of TabItem)()
}
_iUICApp = UIControlledApplication
Tabitems = Tabview controls
tabDockManager = Dockingmanager control.
With this I can get access to the tabs change their style, the dockingmanager is used to update the style if the active one changed.
Note that the TabItem is a control inside the tabcontrol, and there can be multiple tabcontrols!
For changing the docking / position of tabviews a style would need to be changed which one and how the style need to be formatted I don't know. But as this is by now the AvalonDock the answer should/might be in their documentation.
I think in the Avalondock's "LayoutDocumentPaneGroupControl" or the "Dockingmanager" could be a start..
But this will be unsupported and may yield unexpected results in Revit.