It will absolutely be noticeable, because the arranger wanted to have a clear, deep bass line. Moving the line up one octave removes that effect. Now, if you did play your variant, nobody would stand up in the back of the room and shout "YOU'RE PLAYING IT WRONG". But having the deep bass line is fairly important in this music.
You should, if at all possible, avoid changing the arranger's notes. In this case, it means playing the bass note separately from the rest of the chord. Wheat describes how to 'roll' the chord, but in this case, I'd recommend more of a jump; play the bass note an eighth beat early, sustain with the pedal, and then jump to play the rest of the chord on the beat. This avoids altering the rhythm of the melody line, which I personally prefer. Wheat's suggestion will put a little bit of a delay in the melody, which will yield a more schmaltzy (to use the technical term) feel. The choice of which to use depends on the sound you prefer.