I would suggest getting the bags tagged to LHR and collecting them there and storing them at left luggage. 3.5 hours is plenty enough time to collect the bags, store them and get back through security. You should be able to get boarding passes issued the whole way to DXB when you check in for BA122.

You say that you have 2 large carry ons; I take it that that is 1 piece each *and* that they comply with the size restrictions for UK; if not, then store these too and just take a valid sized carry on with what you need for the flight to DXB and back

Dave