That would be OK, but make sure EDI do it properly, with a specially sealed bag and with the receipt, showing location and date, clearly visible. There are some duty free units in LCY, but from superficial examination the choice at EDI is massively better. If you had something specific in mind you could mention it and see if someone here can check if it is available in LCY. Also some LCY shops are very prepared to take a phone call from someone to get a particular item reserved off for you, if you give them a few days notice.

The other option is this: because there is no flight connection, and because checked baggage can be a bit risky at LCY (very little of it goes on), if you have a checked bag exited out at LCY, you can add the item into that bag, then stroll just a few meters to the BA desk for rechecking the bag. It would add very little time due to the precise layout and general efficiency of that airport.