BA are indeed being strict about no through-checking.

However, if you were ever going to persuade them to do it, then transiting through Heathrow between two BA flights would be the one time it'd be logical for them to concede. So just plead with them a little bit, play the sob-story of having to collect your luggage and drag it across to T5 to recheck! Point out that it'll save a job for a BA check-in agent at T5 too

Make sure you have the record-locator and flight number for the second leg written down on a post-it too, to show that you're ready and organised.