You can add following code in your theme’s (child theme’s if any) functions.php

add_filter( 'fep_directory_output', function( $output ){
	$pos = strpos( $output, '<input type="hidden" name="feppage"' );
	if ( $pos !== false ) {
		$output = substr_replace( $output, '<input type="submit" value="Search" />', $pos, 0 );
	}
	return $output;
});

it may break in future.

OR you can override directory.php template to add that button. Instruction to override template in https://www.shamimsplugins.com/docs/front-end-pm-pro/customization-front-end-pm-pro/change-templates/

overriding template works as follows
If in future any template is modified in this plugin and you already overridden that template in your child theme then your template will be used instead of this plugins one. So if any new option or anything added in this template that you will not get until you add that in your template manually.

Currently no plan to change in directory.php but you should keep an eye when update this plugin in future. Most of the time any template change is mentioned in changelog. If is better always read changelog before update (for any plugin)