As an alternative have you checked going SJU to MAD direct and then onwards to LHR, skipping the US entirely? Don’t know your dates but there is certainly Avios availability on the direct flights via Iberia. It’s a bit of a lazier day as the flight doesn’t leave till 6pm-ish but it’s a nice enough island to potter around.

Alternatively change your flight to NYC and get a flight from there, where there’s usually plenty of Avios flights via BA.

Both options depend on being able to do something with the cash flight….