I'm not an e4 player, and I don't play the Sicilian, so maybe my advice isn't the best here.

But I don't see anything blatantly wrong with either move. My preference would be for 4.a4, though. With the following reasoning:

With 4.c4. After 1.e4 c5 2.Nf3 d6 3.Bb5+ Bd7 4.c4 Bxb5 5.cxb5, your c-pawn is no longer covering the centre, so black can take aim against the weakened central dark squares (d4 can no longer be guarded with a pawn) with 5...Nf6 6.Nc3 g6. I think playing slowly as white here is potentially dangerous; if black is allowed some time, the weakness of the d4 square might hurt white. Thus, I think for white to push, active play must be called for, d4 should be played, either immediately, or very soon, and white should gun for black's king. The thing is, with the pawn on b5, black has a clear hook to open up files on the queenside so it seems white's king should go with short castling. This makes it harder to throw white's kingside pawns forward, which makes me feel that this would be weaker for white than a Yugoslav attack against a Sicilian Dragon. It's a sharp fight nonetheless, so there's certainly scope for claiming that white can do well practically if they take the time to study this.

Of course that analysis is very verbose, incredibly speculative, and spoken by a non-expert. But hopefully it provides some food for thought.

With 4.a4, there is no such weakening of the centre squares with 4...Bxb5. Consequently, I think it might be possible to play slowly as white and just play chess. You might well throw away your claims to trying for an advantage in your efforts to reach an original position, but it seems that this is a completely legitimate strategy nowadays.

Personally, I like the idea and would go for it. Not for the purpose of theoretical strength, but for the reason that exploring original positions for yourself is a part of the true romance of chess, and also a sure way to improve your game.