You might have more luck if you plan to go north of the LL zone (not much difference in track miles) and join at BPK to route LAM-DET-DVR. There's a good chance you will get fitted in towards BPK and be above the LTMA departures by the time you get to LAM.
Probably best to see if you can persuade Farnborough or Luton Radar to coordinate your entry with TC rather than getting London FIS to do it (radar to radar vs information to radar).
Good luck.