How to get INSERT errors from $wpdb?

To insert data you should use query() (documentation), get_var() method is for selecting data. If error is encountered, query() returns FALSE.

query()

This function returns an integer value indicating the number of rows affected/selected for SELECT, INSERT, DELETE, UPDATE, etc.

For CREATE, ALTER, TRUNCATE and DROP SQL statements, (which affect whole tables instead of specific rows) this function returns TRUE on success. If a MySQL error is encountered, the function will return FALSE.

$result = $wpdb->query( 'insert query' );
if ( $result === FALSE ) {
    // display error
}