An economical choice, and a timber that's easy to work with, is preservative treated pine. The two main treatments are CCA and ACQ. There has been some concern about using CCA-treated pine for vegetable gardens on account of its arsenic content, and presumably that would also apply to constructing compost bays. However CSIRO considers it is safe with some minor precautions - refer their data sheet titled The facts about CCA-treated timber, available on the net. If preferred, pine treated with arsenic-free ACQ could be used.

The glue joints should not have opened up. Prolonged exposure to direct sun will lower the moisture content of timber, but you report that the treads in shadow are also separating. It sounds as if the timber had a moisture content that was higher than recommended at the time of installation and is now adjusting to surrounding conditions by drying and shrinking a little. Was any specific moisture content mentioned at the time of supply? Even now it would be helpful if the joiner or the timber supplier could test the timber with a moisture meter.

We are not aware of any limitation on the use of mass timber for health projects. In fact a number of health-related buildings have been completed in various countries including Australia, for example here https://www.buildaustralia.com.au/projects/new-landmark-timber-health-and-medical-precinct-opens-in-norwest/. In saying this we are talking about the structure.

In BAL-40 areas 9 mm fibre-cement is a deemed to comply wall cladding on its own, as long as it is backed by sarking on the outside of the house frame - refer Australian Standard 3959:2018, clause 8.4.1 (b) (i). So an additional layer of fire resistant plasterboard is not needed.