Actually they breathe compressed air (not specifically higher in oxygen content than uncompressed air) so that would be an option. I'm not sure why I was thinking that this would not be viable.

However, as CO2 sinks within normal atmosphere compared with smoke which typically rises, as the dry ice sublimates it becomes much more of a hazard for the area surrounding the fire.

For an extreme example of exposure to CO2, see the Lake Nyos disaster [1] that killed people & animals in an area 16 miles around the lake.

[1]https://en.wikipedia.org/wiki/Lake_Nyos_disaster