Not in a similar position, but I wonder if the problem is really susceptible to petition?
It sounds to me like the issue with access to waterbirth in your area right now isn't that the hospitals are necessarily opposed (although they may be opposed, for various reasons, and that's another issue), but that they don't have the infrastructure to support it. If the hospital has only a single inflatable pool, waterbirth will only ever be available to one laboring woman at a time. Pool setup will be time-consuming (especially if it's not something the staff has a lot of practice at), and may be possible only in specific labor and delivery rooms (not all hospital rooms have equivalent floor space, or appropriate plumbing and electrical access in sufficiently convenient places). If the pool is inflatable, it probably wouldn't hold up terribly well to constant use. Draining the pool will be another issue, post-birth, and cleaning both the pool and room will be complicated - it probably adds 2-3 hours to the time the room has to be empty between patients. There may be concerns about spillage, leaks, and whatever's on the floor below the L&D unit. Janitorial staff may have voiced objections.
Basically, if there's only one, inflatable, pool, I think you can petition until their ears fall off, and waterbirth will not get more available, or more advertised, or discussed and offered to more women.
I think you'd have better luck looking into fundraising options for outfitting the L&D unit with bathtubs. The CNM you're working with might be able to tell you what the availability of bathtubs vs. showers is on the unit right now, and whether there are any plans for renovation. I recognize that large, philanthropic outreach projects are infinitely more difficult then sending a petition, but the petition is going to get, at most, a nice note in response. You'll need to find someone to throw money at this problem if you really want it to change.