The function ctools_context_handler_render_handler()
does the trick, e.g.:
First you need to get all handler objects of the given panel, e.g.
$handlers = page_manager_load_sorted_handlers($task, '', TRUE);
Specify your handler (variant), which you want to display, e.g.
$handler = $handlers['node_view_panel_context_3'];
Now we got all needed arguments to fire the function:
$output = ctools_context_handler_render_handler($task,'',$handler, $contexts, array($node->nid));
And now, the $output
contains the variant (3) of my panel.