As of this morning you can add your Known Traveller Number to each BA booking via their web site. At the moment there seems to be a problem adding it to Executive Club member profiles but if you have the patience you should be able to secure PreCheck with BA flights.
In the Advance Passenger Information section, the new box above has appeared.
I will be trying it on my forthcoming JFK-LHR flight.
Great to finally see this – hopefully they’ll sort mobile BPs soon (it works fine with AA so is definitely do-able!)