Bertie - the same suggestions are made every couple of years or so regarding civilianisation of 78 Sqn - the S61s could take the SAR role and a contractor could provide the heavy lift for the times when it is needed.
However, since it is our only deployed SAR asset and the whole military SAR empire is under financial scrutiny from MOD (it doesn't do a whole lot of military stuff despite being a vital part of the nation's obligation to honour the Chicago Convention in providing SAR facilities for shipping and aviation), to hand it over in the current political climate would be less than wise.
Once the future of SAR helicopter provision in UK is clarified, since you currently have 3 service providers - the RAF, RN and HMCG using Bristows, where probably one would suffice, then maybe it will be handed over.