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Unless you are able to redeem for an award that allows for a free stopover in LON en-route to LCA (you might be able to do this on a redemption with Aeroplan or UA or US), for LON-LCA, BA Avios, redeemed for a Reward Flight Saver" award, would provide very good value. Just 10,000 Avios and a copay of about $28 USD covers all surcharges and taxes for an Economy Class ticket. Of course, you would have to be able to find award-seat availability on your desired flight. Both MR and UR points can be converted to BA Avios, and Amex sometimes runs bonus promotions on transfers to BA.
At some point in 2015, USDM will be converted to AAdvantage miles. Currently, the AAdvantage program is all one-way awards, with no free stopovers anywhere. So if you could amass enough USDM before the conversion, that might be the way to go. (And US offers a CLT-LHR nonstop.)
Also, bear in mind that lap infants do not travel for free on most international flights, but are instead charge a percentage (10%?) of the cost of a cash ticket.