I am Gatwick.....
Transponder off in the circuit is recommended practice. At all other times, including when under the CTA, if you have Mode A and Mode C please select them both. Both for the benefit of the TCAS equipped airliners and as further usefull info for ATC. Your level may not be verified, but it is still usefull to see.
As for speaking to Gatwick on the route you mentioned, my advice would be feel free. Sadly with the demise of Dunsfold there's no suitable LARS unit around the S and SE of the Zone. But if you call Gatwick you'll get a FIS and subject to primary workload you may be offered a RIS. Ideally listen out to see how busy it is first, and if it is busy when you call keep the r/t as crisp and, err, um, ahhh concise as you can
WF.