Your route probably won't look like that.
Normal departure from Biggin is 21. Used to be with left turn at 2nm. Now they have some nonsense with left turn at 3nm.
So, suggest one of the following:
Start with Biggin. After turning left at 3nm, give Gatwick a call as you infringe their zone.
Or, confirm left turn 3nm with Biggin, but at 2.5 call them back and say "leaving the ATZ, squawking 7000 and changing en-route", then turn immediate left.
Other than that, turn the volume down, call Biggin for rejoin. No need to talk to anyone.
Leeds Castle aren't keen on low overflights, it upsets the birds I'm told and Headcorn get the blame. Just bear that one in mind.