Probably not. Good chance that the doorbell is fed from a Class 2 power supply which means the conductors must be separate regardless of insulation value.
What if i had 240 v to ground and 120 v to ground with same wire 300v and 600v rated?
But be careful to account for the fact that the insulation must be good for the maximum line-to-line voltage, not just the line to ground voltage. So the result will be different between the 240 being 120-0-120 or 0-240 or 240 3-phase delta.As long as the highest voltage is lower than the lowest insulation rating you are OK.