Actually, if White responds by playing c3, Black is actually helping White to develop with tempo. In Ruy Lopez, c3 followed by d4 is White's common reply to challenge the center. So, soon or later c3 will get played.
Odd Ruy Lopez Variation

It does indeed exist. Has a poor reputation but it's the sort of line which attracts a few devote followers who might be able to score well with it thanks to theyr extensive experience. It does not look particularly logical to me to provoke c3, a move which white wants to play anyway, but players stronger than I am have seen something in this line and so there must be some subtler point behind it. Maybe black wants to arrange a quick d5, since c3 is not a move white necessarily wants to play in the relulting structures. 4.c3 Ba5 5.O-O Ne7 6.d4 exd4 7.cxd4 d5 for example looks very reasonable for black. But 6.Na3 intending Nc4 going against the wandering bishop scores well in the database i looked in. Nothing dramatic but white looks certainly more comfortable to me.


I know that exclams are for free, but it's not a good reason to waste them ;) c3 can be played either immediately or after O-O, i doubt the difference is great. However your proposed 4.c3 Ba5 5.d4?! is hardly great for white (the c3 pawn is pinned). I also doubt black will find himself in particular trouble after 5.Bxc6. Just 5.O-O stepping out of the pin is almost certainly best at move 5.

Better, but maybe still not great. This leads us to the line i already mentioned 1.e4 e5 2.Nf3 Nc6 3.Bb5 Bb4 4.c3 Ba5 5.O-O Ne7 6.d4 exd4 7.exd4 d5 which looks rather balanced to me. in fact black has a plus score here in the database i use; white d4 pawn might prove weak.
I think 6.Na3!-Nc4 is better, the resulting positions looks rather balanced if not for white having the bishop pair in an open position. This must count for something(nothing dramatic of course).
I most definitely have not invented this variation. I mean, I came up with it, but knowing the RL it's been analysed so much someone, somewhere, has came up with it first.
Black's last move pins the d-pawn, preventing a future d2-d4 to break open the centre or seize control of it, as well as developing the bishop. A delayed version, playing a6 first, is also feasible.