How can I change default timer ringtone
On a note 4, you can call on the menu button by long pressing the recent apps button on the bottom left of your phone.

(source: here, and personal experience)

So just go into the clock app, go to the timer tab, and long press on the recent button. You should now have the menu that you're talking about. If it doesn't come up, it's most probably because Samsung disabled the option.