C1 Flown Cover June 11, 1927

I have this cover dated June 11, 1927, postmark Washington DC, with special stamp: Lindbergh Day, Special Flight Washington to New York. As I understand Charles Lindbergh flew from New York to Washinggton DC to receive the Distinguished Flying Cross from President Calvin Coolidge. Thereby making Lindberg the first aviator to receive the DFC. My question is do u think this cover flew with Lindbergh on the trip to Washington?

Your post has several errors of fact.
He came back via boat which docked at Washington DC on June 11.
Your cover flew from DC to NYC on June 11.
He flew from DC to NYC on June 13.
Thus he did not fly your letter, or even with it.

The American Airmail Society catalogs cover Lindbergh quite thoroughly.

From AAMC
Lindberghiana, Page 515 & 516, Vol 3, 4th Edition.On June 9th the Dept announced that a special plane would be available on the 11th for dispatch from Washington at 4:30 PM to NY in time to make connections at Hadley Field at 6:30 PM with a special air mail dispatch which would attempt to reach San Francisco by Sun-down on the 12th after transfer at Chicago early Sunday morning.

Lindberghiana-29 June 11 4:30 PM Washington to New York with official 'Lindbergh Day' cachet.
Most cachets violet. 150 black, 1200 magenta. $6 $10 (1974 4th edition, I believe)