seriously i think if u try it out personally, i had this strong feeling that the mushroom can really last very long before it start to melt away... dun even need to leave it under sun. maybe just a indirect light source from ur house lighting will do.
it will be better if there is a way to circulate the water without heating the water. the mushroom will not thrieve, but i think it can really survive real long.
on a side note, i always think the mushroom or coral can photosynthesize and release oxygen because they contain algae.
Extracted from site:Through photosynthesis, zooxanthellae convert carbon dioxide and water into oxygen and carbohydrates. The coral polyp uses carbohydrates as a nutrient. The polyp also uses oxygen for respiration and in turns, returns carbon dioxide to the zooxanthella. Through this exchange, coral saves energy that would otherwise be used to eliminate the carbon dioxide.
http://www.seaworld.org/infobooks/Coral/adaptcr.html