Sorry, not trying to scare you, but is is important.
First start by avoiding the "not in my house" there can be many reasons, incliding the child not talking with you, the child saying it to mom for attention. Instead offer to keep an eye on it and let mom know. See if you are defensive you can't ask mom not to be.
Secondly talk with the child! Ask her about it, open the communication with her.
and lastly, offer to take the child to the doctor yourself! if you live in the same town you should know her primary doctor. Tell the doctor about what is happening ALL of it, including mom's concerns, cause like it or not she may be right. Let the doctor run the test and sort it out.
If you don't know her doctor, GET to know him, make sure is up to date and understands the family situation, but try to be gentle about explaining it, and don't do it in front of the kid. you know instead of "mom is a hypochondriac, mom has a tendency to jump quickly to the worse case scenario, she worries about her health"
BTW to be on blood pressure meds takes a doctor script, and while in the pyschology world it is easy to say the right things to get kids meds, in the medical world, the child would actually need to have low blood pressure.
One other thing, change your perspective, you are griping that mom TOLD you. Heck many of us would give our souls to even be TOLD about what is happening with our kids.
you say mom doesn't listen, but she is changing her diet so maybe she is thinking it is not just breakfast but the difference in foods.
On the rights issue, it depends on your state and the custody agreement. If mom has 100% sole and legal custody, then technically you can do nothing.
If you have joint legal, some states define that as the right to KNOW, others that they must be made "jointly" but as medical fields only require the permission of one parent, they are not liable for the parents communicating and deciding.