Since there is not much play here, I will state of obvious first, while enjoying an outdoor activity I feel your runs would be included into the following RCW as a lawful outdoor recreational activity.
RCW 9.41.060Secondly how is carrying ID for driving, credit or debit cards for purchases any less cumbersome then carrying your CPL?
Exceptions to restrictions on carrying firearms.
(8) Any person engaging in a lawful outdoor recreational activity such as hunting, fishing, camping, hiking, or horseback riding, only if, considering all of the attendant circumstances, including but not limited to whether the person has a valid hunting or fishing license, it is reasonable to conclude that the person is participating in lawful outdoor activities or is traveling to or from a legitimate outdoor recreation area;
Thirdly carrying ID is a good thing especially when outdoors running where unfortunately bad things can and do happen, thus why you carry. Something occurs and you are unable to tell them who you are, ID could save your life by finding out medical information quickly as possible.
If it is on this issue of being with out identification or CPL is ridiculous, if you do not want to show ID to an Officer when requested, then tell them that.