The first routeing then:
MID-BIG would take you through the climb out/final at Gatwick.
OCK-BIG line is not separated form Heathrow departures. 3A, 4A, 5A, 6A are all out. FL70-120 are use by Heathrow inbounds.
BIG-LAM has City/Biggin traffic at 3A and 4A - and Gatwick SIDs and City departures at 5A. Heathrow SIDs go there at 6000ft. FLs mean in the stacks again.
CPT - BNN - BPK as suggested soesn't work if there are BPK SIDs off City, or CLN SIDs off Luton (and some Northolt SIDs) 3A,4-5A are blocked.... and then the Heathrow departures again climb to 6A in that area. Above that are FLs for Heathrow.
So - what normally happens is that traffic into Elstree/Stapleford/Denham arriving from the North East - South East end up working Thames via DET or SPEAR at 4000ft. However, they are normally given descent out of CAS if there are Biggin/City traffic around.
Most biz-jets positioning within LTMA airfields have company procedures that do not allow transits outside CAS. In the last week I have seen a Northolt to City go on a CPT SID - route South of Gatwick to come in at DET! (FL160 I have seen!) - I have also seen the same operator go City - Farnborough on a CPT SID.
The best way is to file a SID that gets near to a STAR near your arrival airfield. (Unless of course you were planning to do this at 0300hrs!!!)