All the MOD SAR force is technically deployable I think. But concur that the RN rotates the crews way more than the light blue.
One of the fundamental difficulties for the civ's (I standby to be corrected here...) is that they remain bound by whatever limits the MCA or whoever sets them. Whereas a mil crew can keep going at the a/c cdr's discretion with no theoretical limit, merely judgment as when to say "no". Not that the civ's would not have the cojones to keep going (they're mostly ex-mil anyway) but for them the line is the line. Mainly I'm talking about actually getting to the scene, which is often the hardest bit.
Last autumn a mil crew had a bugger of a time getting to a family of 8 who were literally being washed away on a river island (their transit van already had). The wx was diabolical but by climbing over the trees when necessary, hover taxiing down the roads etc, they got there and pulled them off, in the nick of time when things were looking v.grim.
How did the crew justify it? Because it's operational flying which allows a/c cdrs to modify the rules if it's in the national/service interest.
How can such rules be applied to a civ crew? Bristows/CHC or whoever have a ton of money invested in their cabs - they won't allow them to fly "limitless".
Perhaps the answer is for the MOD to "parent" all SAR and take the budget that would have been allocated to the MCA post SAR(H). This is of course a great theory but the reality is still likely to end up way different I'm afraid.