The Mafia in this country (if not with the big M...certainly organized crime with a small m then...) tried the same thing in the garbage collection business. They would approach the local governments with ridulously low prices that undercut the local units costs by a bunch.....get the contract....hold the prices down until the government unit shed itself of all of its equipment and personnel....then the prices headed up quickly. The strategy was to keep the new prices well above the original government operation costs....but not so high the government could purchase a whole fleet of equipment and hire new personnel thus the taxpayer would lose and "organized" criminals would benefit.
Not saying that is the case in this situation but a similar situation could occur just as you describe. I would think honest competition by the various operators would work to defeat that concern however. Instead of the RAF/RN competing with private business....the competition would be between the private operators.
The RAF/RN would have to maintain a Combat SAR capability along with their other committments thus that expertise would not disappear from the RAF/RN skill sets by non-combat SAR going to the civvies in my view. I could see the military SAR units being a backup to the civvie SAR for domestic operations.