I tried what you suggested and several other things, and still cannot get a timer without the ringer turned on and "Do not disturb" turned off.
I put the device on "Do not disturb" with the sound on, and then created an "Exception" within "Do not disturb" for the Clock App. That did not solve the problem. The timer refuses to go off with "Do not distrub" turned on. I prefer "Mute" rather than "Do not disturb", because my phone still makes clicking noises when I type and "Do not disturb" is on.
I also tried downloading two different timers when "Do not disturb" was on (but phone sound was on), and neither made sound, even when I had those apps as exceptions under "Do not disturb".
Any other suggestions? I find it hard to believe that the only way to use a timer is to turn the ringer on. There are many situations (work, timer for a nap, etc) where someone wants to use a timer on a phone without getting calls.