... you can book a flight and car package with us anywhere on the OneWorld network.
The problem with that is that the majority of PAX getting off at the destination end up hiring a car from the company paired with the airline. That leads to 'effin big queues at Hertz, and no-one at the Avis/Europcar/Alamo desks. So I would always book a car from a rental company NOT paired with the airline I arrived on.
Also BA are the only airline that have ever bumped me off - twice in fact. EJ and Ryanair never have, and as long as you play by their rules they'll get you there. So I avoid BA as best as I can, although sometimes I'm forced to uses them due to route or time constraints.
We have a corporate code for Hertz, but our primary Car rental tie in is with Avis and Budget. I get three different ways to book, only one of which is retail, and will research the prices ahead and book whichever is preferred the day I fly, ie when I know for certain the flight is operating... (You'd like to think that there is certainty with this, but I have worked on the frontline long enough to know better... It blows my mind how many of my colleagues get all upset about relatively minor disruption in an industry where 24 hour delays were once routine). I would never wait until I landed to get a car unless it was an emergency.
There are several potential reasons for getting bumped. It's not always the result of overselling that causes it, but disruption (not always immediately obvious as it could be elsewhere in the network), last minute aircraft changes where a flight is sold on one variant but gets switched to another due to an issue impacting the original aircraft (again this could be environmental issues elsewhere such as weather where the original aircraft was coming in from rather than a technical fault) as our 787s are being refitted they have fewer business seats which sees people getting bumped down a cabin, and it's quite often the lowest fare paid that gets knocked out of the back to make space. Broken/locked out seats are another cause of this. A large number of delayed connecting passengers can impact availability on subsequent flights, as can missing your own connection.
Booking a cheap return flight to somewhere like Copenhagen and then book your trans Atlantic flight from there may offer some extra bump protection and actually be cheaper than flying direct from Heathrow, just be sure to allow enough leeway for your subsequent connections. Again booking direct rather than through an agency affords greater protection because it makes you the customer of the airline rather than a customer of the agenct who in turn becomes the customer of the airline. Doing this should also protect your connections by not allowing you to book a ten minute connection

I have seen the impact of this first hand where flights get changed for any number of reasons and the "customer" ie the agency that sold the passenger the ticket gets informed of the changes but fails to pass this on to the passenger. Had they booked directly, they would have been updated directly.
I know BA isn't perfect, I work for them inspite of that, and I quite often fly with The Orange Product because that particular route works better for me. I have never, and will never fly with Spud-U-Like for several reasons, none of them positive.