Can I leave the cushion in the bath

You can leave the cushion in the bath at your own discretion.
If the cushion is left in direct sunlight, it is likely that it will discolour (fade yellow). 
Some oils/emollients that are used in the bath along with the Bath Buddy, can also cause discolouration but this is not often permanent. This can be cleaned up with furniture polish to restore back to its original colour.