South West Midwifery Practice has just launched in Bunbury (on Jan 16th, so it's still very new!). It allows eligible low-risk women to choose home births.

(Hopefully it's okay to quote the contact details?? Apologies if I'm not meant to do this.)

Women can apply to be part of the program by directing self-referral enquiries to SCHS-SWBHMidwiferyGroupPractice@health.wa.gov.au until January 25 and to WACHS-SWBunburyMGP@health.wa.gov.au after January 25.
In regards to Bunbury Regional Hospital - they do have 2 (or is it 3??) big baths. They are not in any of the birthing suites though, they are across the hall.