Or.... Do nothing.
Calm down.
So 2 medicare cards that is because the kid has two families. They are not linked so Medicare ain't gonna help you.
So even IF you managed to find out what medical practice the kid went to, you're not gonna be able to get anything out of the doctor who saw the kid.
Story time - my ex didn't believe me when I told her we were all at risk of having whooping cough. My partner and 2 of the kids that live in my house were confirmed as whooping cough. Bloody terrible illness btw - get immunised. . Meanwhile I was given a week off work because I was a risk and I could potentially have passed on the bloody thing. So we have a week off school / work and directed to minimise contact with the outside world. So me and the two boys have no symptoms, fit as a fiddle. So we're off to the skate park during the day, going fishing. But staying well away from the rest of the world. It was great... But, the ex doesn't believe me. Thinks I'm taking the kids out of school for fun. Demands to speak to the doctor. The doctor would only talk to the ex with my consent (which I gave). Ex calls doctor, he confirms whooping cough diagnosed in 2 of us BUT he refuses to discuss anything else. WHY? because he doesn't have to and that is all I gave him permission to discuss.
Yes, your ex should be telling you about doctor's appointments - that is covered in the shared parental responsibility part of the legislation. But medicare, DHS or the medical practice will not be telling you anything without a court order that forces them to do it.
NEXT - the kid had a sore throat? not a viscetomy? Pick your battles - your blood pressure is going up because the ex wont tell you about a common childhood illness. Sure, absolutely bad form on her part - but if you loose your sh*t over it SHE WINS. Pick your battles and this is one that is not worth fighting. She is withholding information to piss you off. It has worked, she has won.
BTW - the kid was sick - the kid saw a doctor - so what is the problem? NOTHING - Nope, your ex is the problem but there isn't a doctor that can fix that, True...
Now go and take the dog for a nice long walk and calm down...