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"Is there a way to get a layover in LHR?"

"Layover" is an ambiguous term. On international itineraries, a layover of more than 24 hours is a stopover; a layover of less than 24 hours is a mere connection, which AAdvantage rules permit.

If you want a true stopover in LON, acquire some BA Avios, and redeem them for a cheap "Reward Flight Saver" redemption from LON to elsewhere in Europe.