How can I perform programmatic routing without a URL change?

You can perform routing without changing the URL with the help of NavigationManager. You need to inject NavigationManager to use this service. You can use this service to perform routing programmatically.

Service Injection

@inject NavigationManager Navigate

<button @onclick="NavigateTo">Navigate</button>

@code {

      private void NavigateTo(){
             Navigate.NavigateTo("url");//specify required url here
      }
}