First, before the interpreting vs. interpretation issue; the articles are missing in the first sentence and I think and should be at. It should probably be the definitive article:
I am applying for the interpretation position at the Department of Defense.
Or if the job is not well defined:
I am applying for an interpretation position at the Department of Defense.
The interpreting vs. interpretation issue can be avoided by using a job title instead:
I am applying for the position of Interpreter at the Department of Defense.
or
I am applying for the position as an Interpreter at the Department of Defense.
Second, experience in this context is usually not counted, like in Last year I had four car accident experiences, but a matter of degree - little, extensive, fair amount of, a lot of.
It could be:
With three years of professional experience with interpretation and translation for Fortune 500 clients, I believe I could contribute to the Department."