Black's main threat after 3... c6 is d5; it looks like Qe2 is an attempt to prevent that, by indirectly attacking pawn e5, even with check. However, I'm not so sure that it works as intended, because Black can play d5 anyway, e.g.
1. e4 e5 2. Bc4 Nf6 3. d3 c6 4. Qe2 d5! 5. exd5 cxd5 6. Bb5+ Nc6 7. Qxe5+ Be7 8. Nf3 O-O 9. Bxc6 bxc6 10. O-O Bd6
The Bishop's Opening often transposes to the Italian, so black's response depends on which line he wants to play after 1.e4 e5 2.Nf3 Nc6 3.Bc4: 3....Bc5 or 3....Nf6. The former allows the Evans Gambit (4.b4), while in case of the latter, black should be ready for 4.Ng5.
According to the GameDatabase of ChessTempo, after 1.e4 e5 2.Bc4, the most popular ...
5...Bb4+ looks like a weird move because you wouldn't play a move like that often, but it makes perfect sense because the white bishop in on b3.
White would definitely want to develop his knight with Nc3 on the next move and put pressure on Black's pawn center. By playing 5...Bb4+ you prevent 6.Nc3 (it'd lose a piece). If White goes for 6.c3, then Black can ...
3...c6 is in my opinion by far the best move here. Usually white would continue with something like 4.Nf3 Be7 5.O-O (5.Nxe5 Qa5+) d6. 5...d6 is a more solid choice but 5...d5 is another option if you want to blast open the center quicker. For a nice game with 5...d6 played, take a careful look through Gelfand - Yusupov, Munich 1994.
Black cannot play Nc6 at the moment, that is true. However, it doesn't look like White can exploit this immediately, either.
Black will answer 3 Nf3 by ...d5, gaining tempo on the Bc4 that he can then use to either defend e5 (d7 is open for the knight, at the very least) or counter-pressure e4 (which is equally weak as e5). The whole motive is similar to ...
I have played this in the past, and always had trouble with the following line:
1. e4 e5
2. Bc4 Nf6
3. d3 c6
4. Nc3 d5
You can also play Bc5 on move 2, to prevent the Urusov Gambit, 3. d4, and likely transpose either into the Italian Game, or white may continue as above into a King's Gambit Declined setup.
You can go and convert it into a Giuoco Piano chess opening with the moves:
1. e4 e5
2. Bc4 Bc5
3. Nf3 Nc6
Now you have transposed into a regular opening line and can carry forward from there.
Usually 2...Nf6 is played and only then 3...c6; this makes a difference.
2...c6 is rarely played and in response 3.Qe2 is strong.
With 2...Nf6 White is encouraged to defend his e-pawn. If 3.Qe2 Black can switch from 3...c6 to 3...Nc6/3...d6 lines and the Queen on e2 doesn't help ideas of playing for d2-d4. Now with 1.e4 e5 2.Bc4 c6 3.Qe2 Nf6, White can ...
2...c6 is one of Black's best replies to 2.Bc4, as it aims to play ...d5 and take advantage of White's bishop being developed so early. If White let's this happen, Black gains a big centre with tempo, and White loses any chances for an opening advantage.
As you noted though, the weakness of ...c6 is that Black cannot play 2...Nc6. Thus, the best way to play ...
I would play d5 in this position. It attacks white's center and gives you a lead in development. The only problem is that it's the reverse Scotch Opening, which means the more aggressive Max Lange Attack isn't possible.