Why don’t you tell employee that?は伝わると思いますが
Why don't you tell an employee?の方が自然な言い方です。
employeeという名詞の前、a/anは必要です。
又、thatを最後につけなくてもいいです。
「tell someone that 〇〇」はよく使う表現です。例えばWhy don't you tell an employee that it's brokenとか。
〇〇してみれば?
Why don't you try 〇〇ing?
Maybe you could 〇〇
Could you try 〇〇?
ご参考になれば幸いです。