If you were to fly into Heathrow and book the return flight to Amsterdam from Gatwick then British Airways will only check your bags to Heathrow. As it is no longer allowed, or even possible, to through check-in baggage with British Airways when there is a ground transfer between Heathrow and Gatwick or verses versa. i.e. AMS-LHR-LAX/SAN-LHR-LGW-AMS
There may be a small difference in air fares as the LHRAMS sectors are usually more expensive than LGW AMS. Also you need to make sure the connection time between arrival at LHR and departure from LGW is less than 24 hours. To avoid the UK departure tax becoming applicable and a higher grade fare being calculated for the whole journey.
For your London shopping trips you could try the same in the opposite direction i.e. MCO-LGW-Shopping-LHR-AMS Although you would have to take your luggage with you into London so it was really available to take onto LHR. Not ideal, but something to consider.