My DB shows one game that follows this line a ways. Black resigned immediately after 7.Nd5.
Damiano Defense (3...g6)

W can seek a certain edge also by simply letting the pawn go with 5.Be2 Qxe4 followed soon or later by Nc3. I don't think 3...g6 improves on, for example, 3...Qe7; it looks like it merely gives W the extra option or trying to keep the pawn.
Thanks anyway for posting your idea!
We all the the Damiano is a piece of manure. W had all learned that either from our own games or from amusing ourselves with Greco's miniatures.
However, I have been taking a look at a gambit line in which Black has a hypermodern attack on the e4-pawn.
My analyisis is most likely faulty, but I just thought I would throw it out there.