He'd have difficulty advancing in either party.
Bryan was heavily involved in the fundamentalist-modernist controversy and was loudly anti-evolution and pro-prohibition. Even updating his social views to their modern equivalents, he'd still be hopelessly out of line with today's Democratic Party. Obviously his economics wouldn't fly with the Republicans either.
If he got into politics at all, he'd be some eccentric state representative that we'd only hear about in a hysterical U.S. General Discussion thread.
I think if he managed to make it in the House, he could keep winning election like Walter Jones and Jim Traficant. Perhaps he could manage a deal with whichever party he belongs to to become Speaker.
His oratorical skills alone would be reason enough not to mess with him.