I sympathise with your logic... However, Jigsaw is a privately funded service sponsored by BP as a supposed substitute for some of the rescue facility offered by the rig support ships. It was originally to be much more extensive at one point but withered after much political and union dissent within the oil industry. Then the money ran out.
It would be nice to think that some sense would prevail along the lines you suggest. To most it seems a waste that 2 very capable SAR helos sit on standby at the same base , to do the same job - and neither is exactly overworked operationally. However thats politics for you... and the divide between public and private. I suspect the GAP contract of which you post is not allowing such a sensible compromise.... It certainly wasn't considered previously for the SAR-H contract either.