This is not something which you can easily hide from BA and it is something which air carriers are on the alert for as the consequences of a medical diversion over the Atlantic are not inconsequential.

Not only does your own physician need to know that you wish to fly, but she needs to know the exact nature of the flight and may also consult with BA's medical team (contractors) if high-altitude long-haul is not within her specialty.

As noted above, this is not about trying to dupe BA because what is in BA's interest is also likely in your interest as well. What worked for one patient may or may not work for you. That's why people go to medical school.