This is not officially documented but you can implement the IWaitCursorManager interface, here is its definition (in C#):

[Guid("fc992f1f-debb-4596-b355-50c7a6dd1222")]
public interface IWaitCursorManager
{
    HRESULT Start(CURSORID id);
    HRESULT Restore();
    HRESULT Stop(CURSORID id);
}

public enum CURSORID
{
    CID_WAIT = 0,
    CID_APPSTARTING = 1,
}

To implement it you must pass a "site" to the IContextMenu implementation. For that just QueryInterface it for the IObjectWithSite interface, something like this (again in C#)

var ows = (IObjectWithSite)contextMenu;

and then pass a COM object of yours that must implement IServiceProvider:

var mySite = new MySite();
ows.SetSite(mySite);

Your COM object will be called on QueryService for a IWaitCursorManager's IID (among other IIDs). Pass an implementation of it that does nothing, like this:

public class MySite : IServiceProvider, IWaitCursorManager
{
    HRESULT QueryService(Guid siid,  Guid iid,  out object obj)
    {
       if (iid == typeof(IWaitCursorManager).Guid)
       {
          obj = this;
          return 0;
       }
          
       ...
    }

    HRESULT Start(CURSORID id)
    {
        return 0;
    }   
    
    HRESULT Restore()
    {
        return 0;
    }   

    HRESULT Stop(CURSORID id)
    {
        return 0;
    }   
}