well my 2 comments got upvoted, so who knows, maybe i can turn this into an answer.
(Note: I wasn't really into chess or chess960 at the time I read the book (except that I played a few tournaments as a kid), so wow this was a little nostalgic. When I did get more into chess/chess960, I read about the silman position. That's about it.)
jeremy silman talked about the position in these 2 links in the description here. unfortunately, they're both dead. let's see if i can dig up web archive. oh yeah here we go
(also http://web.archive.org/web/20170801000000*/jeremysilman.com/shop/pc/Creating-the-Harry-Potter-Chess-Position-87p3692.htm and http://web.archive.org/web/20111201000000*/jeremysilman.com/movies_tv_js/harry_potter.html)
aaaand of course there's agadmator: Ron Weasley Sacrifices Himself for the Team
anyway, let's see if the game in the movie satisfies the conditions
White starts with 1. e4
- Yes. Ron does Scandinavian defense.
The white queen takes both black knights
Well, no, just Ron. But the other knight doesn't really do anything, so I don't see why there couldn't have been a position where Harry, Ron and Hermione have lost their other knight.
Wait, maybe, there was AN EARLY UNDERPROMOTION.
The second black knight (Ron) sacrificed himself to the white queen in order for a bishop (Harry) to travel 3 spaces to the left for mate
- Yes, assuming you don't count the intermediate move of capturing the queen (the movie didn't seem to count this!)
The first black knight was taken for a trade for a black rook (Hermione) to take a white bishop
- see (2). Also, Hermione doesn't seem to have moved much. But she does have control of the f-file.
Both the bishop and rook (Harry and Hermione) were in danger twice
Harry: Yes. Immediately before Ron checks king at end, Harry, as the dark square black bishop, is in line of sight of white queen.
Hermione: Seems possible. There's no indication for Hermione in danger, but I don't see why there can't be a game like this.
The second black knight (Ron) took the most white pieces
- When playing a game, I don't know of anything where people keep track of which piece captured the most pieces. This sounds something like determining which piece is like the 'MVP', like the online pokemon TCG or csgo. I don't think this is a chess thing. But I guess it seems plausible.
The same black bishop (Harry) went 4 spaces to the right in the opening
- Harry as the dark square black bishop could've done Bb4, like in Nimzo-Indian Defence, but Nimzo-Indian Defence is a response to 1.d4 and not 1.e4. Perhaps there's some variation of Scandinavian that involves Bb4. According to this, there's something called 'Scandinavian, Anderssen counter-attack orthodox attack' that goes
- e4 d5 2. exd5 Qxd5 3. Nc3 Qa5 4. d4 e5 5. dxe5 Bb4 6. Bd2 Nc6 7. Nf3
[White "Some 1.e4 player"]
[Black "Some Scandinavian player"]
1. e4 d5 2. exd5 Qxd5 3. Nc3 Qa5 4. d4 e5 5. dxe5 Bb4 6. Bd2 Nc6 7. Nf3
Also, notice that before capturing the queen, Harry is along the same diagonal of b4 and Harry's starting square (f8).
- But actually 4 spaces to the right is a pretty common thing I think like Ruy Lopez.
Update: Does Magnus Carlsen Remember the Position in Harry Potter Movie?
Here, Carlsen claims that the moves Qc3 and Nc6 were played. Not sure either of those moves were played, but in the silman position Qxd3 (not Qd3 and not Qc3) and Nh3 (not Nh6) were played.