You might want to check with the National Park Service who administrate the memorial. They have a regulations on video
As far as I know it is open to public and in an outside setting. I'm not sure if you would need to get that approved by anyone.
Apply for a permit just like Hollywood does
Park Service or the city
I don't believe you can "rent time" at the memorial since it's a public space paid for by your tax dollars. On a week day in the winter it won't be busy.
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