drop-down list

Yes, but it takes some coding. You need a filter on relevanssi_fuzzy_query that modifies the query based on the dropdown selection.

add_filter( 'relevanssi_fuzzy_query', 'rlv_dropdown' );
function rlv_dropdown( $query ) {
  if ( isset( $_REQUEST['search_type'] ) && 'startswith' === $_REQUEST['search_type'] ) {
    return "(relevanssi.term LIKE '#term#%') ";
  }
  if ( isset( $_REQUEST['search_type'] ) && 'equals' === $_REQUEST['search_type'] ) {
    return "(relevanssi.term LIKE '#term#') ";
  }
  if ( isset( $_REQUEST['search_type'] ) && 'contains' === $_REQUEST['search_type'] ) {
    return "(relevanssi.term LIKE '%#term#%') ";
  }
}

That should do the trick, with a dropdown like this:

<select name="search_type">
  <option value="startswith">Starts with</option>
  <option value="equals">Equals</option>
  <option value="contains">Contains</option>
</select>