FreeRepublic - The "Ruy Lopez" line in your OP is covered in a Chessable course, Arjun Kalyan's Lethal Italian, which is a White repertoire based on the Italian. Kalyan follows your line all the way through 13.h3, which he describes as a novelty in OTB chess that has appeared in correspondence play. He goes on to cover 13...Rb8, but most of his attention is directed toward your idea of 13...Qe8, which Kalyan meets with 14.b4. After 14...cxb4 15.cxb4 d5 16.Nc2 Rd8 17.b5 dxe4, the preview on Chessable ends, though it appears that Kalyan's line continues. Stockfish evaluates the position after 17...dxe4 as equal.

Chessable's position search feature indicates that your line up to 11...fxe6 is covered from Black's POV in the newly released Wesley So's Elite 1.e4 e5. However, an examination of the lines in So's course seems to show that he recommends 7...Be6 instead of 7...Na5. I'm not sure if the position search is in error or if So's coverage of your line simply can't be seen because of the limited previews provided by Chessable.

Dubov I think is a ...Be7 player, so look at his games. That may provide you an insta-repertoire of sorts.

I remember a GM comment about d3 Bc5 to the effect that he did not understand all the nuances, but decided that no one else did either. The same may apply to  to d3 Be7. There are annotated games in ChessPublishing. When it comes to a recommended repertoire for black there is Lokander's Opening Repertoire, previously mentioned. That's all I'm aware of. One of his lines goes

1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 Be7 5. O-O O-O 6. Nbd2 d6 7. c3 Na5 8.
Bb5 a6 9. Ba4 c5 10. Re1 b5 11. Bc2. My note is that this has transposed to the Pilnik system of the Ruy Lopez:
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O! Be7 6. Re1 b5 7. Bb3 d6 8. c3
O-O 9. d3 Na5 10. Bc2 c5 11. Nbd2.
I find it remarkable that two very different move orders arrive at the exact same position on tempo. I'm not aware of a way for either side to steal a tempo.

I could not find a forum entry on 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 Be7. It's a main line, second only to 4...Bc5 in popularity. 4...Be7 was recommended by Martin Lokander:

https://forwardchess.com/product/opening-repertoire:-the-open-games-with-black?s...

The play can continue 5. O-O O-O 6. Re1 d6 7. a4. At this point Lokander prefers 7...Kh8 to the more frequently played 7...Na5.

I am curious about treating this like a Chigorin Ruy Lopez. In my notes I have 7...Na5 8. Ba2 c5 9. Na3 Nc6 10. c3 Be6 11. Be6 fe6 "slightly more pleasant for White." Continuing on a little, 12. Bd2 a6 13. h3, black seems to have good chances in an unbalanced game (13..h3, 13...Qe8, 13...Rc8).