I agree with you about this seeming too coincidental, but I'm not a medical professional so this is just a gut feeling. If it were me, I would insist on urine and blood cultures. Sometimes UTIs can present without the typical symptoms or burning, etc. Is the heart murmur you mentioned been since birth?
The fact that the fever returns at the same time with each antibiotic start-up is interesting. Do you keep a journal of symptoms, fever, general activities, start-up of antibiotics, etc.? If not, you might consider a daily journal recording these things – even on good days.
If the fevers continue to be a problem, I would seek a second opinion with a well-known children's hospital in your area, such as a hospital connected with a university medical school or a Mayo facility.
If you are comfortable doing so, will you keep in touch and provide updates?