No. You just have to play the notes that are below your low E up an octave. It works fine. You don't have to play the whole part up an octave. People are all different - their voices do not all function best in the same key(s). Phil Starr sums it up very well above. I repeat - if the singer is struggling, a song will not sound good, no matter how well the rest of the band plays it. As musicians, we are there to support the singer and the song, not to indulge ourselves or show how clever we are. And if you do pro work, you had better be ready to play a piece in the key required, not the one you think it should be in.
I'll give an example. A band I used to play in did 'Grapevine'. Our singer, who was (and still is) very good, could not manage it in the original key. It was just too high for him, so we dropped it to one that was comfortable for him. He was in his fifties. Marvin Gaye was the best part of 30 years younger than that when he recorded the song and we lose our upper register as we age.
The guitar player kicked up a terrible fuss about it. The real issue was that he didn't want to learn it in a different key. He would rather the song had sounded poor than make an effort. A bit pathetic, really.