Without having checked against a current chart:
I used to transit this area regularly, and we often went IFR at 3,000 or 4,000, because it's busy in the FIR. But:
You could:
Report leaving Biggin's ATZ
Call Biggin for rejoin (somewhere around Sevenoaks is sensible)
I would probably remain below 2,500 ft QNH and:
Talk to Biggin (obviously), until leaving their ATZ, at least.
Talk to Farnborough (as suggested above)
Keep a little further north of Rochester (getting clearance into Southend if you go far enough north for it) or talk to Rochester from slightly east of Gravesend to north abeam Farthing Corner (especially in there is any south in the wind) and then talk to Farnborough
Talk to Biggin
But MOST IMPORTANTLY IN THAT PART OF THE WORLD: look out diligently, and use TCAS if you have it.
if you have requested a Basic Service from Farnborough, they would keep you informed of other traffic if not too busy on the frequency
Emphatically, no: they 'might', perhaps, but certainly not 'would'. If you want a traffic service, ask for it.