I'm not a security expert. However, this part looks worrying:

> alice can also hide messages from times she wants to keep private

If there's a need for this, doesn't that imply that the scheme does not actually keep Alice's privacy in all situations?

Furthermore:

> the random messages give the hospital NO INFO on where Alice was

This seems to assume that the hospital (or anyone with access to the data, such as governments) didn't capture the broadcast messages together with their location. With enough Bluetooth receptors in busy areas, a government could easily find out where Alice had been by looking up each of her messages in their list of message/location pairs?

Experts can probably come up with nastier and/or easier exploits...