Please don’t consider the below legal advise. But I think there are three issues here:
Is the damage caused by publication (of the patent) offset by the potential to obtain money by licensing/suing
How should the claim be drafted
How does one detect infringement in “hidden” cases.
Re 1. I would ask if you are sure that only that three-letter entity will infringe. Otherwise, there may be more value in maintaining (and selling) as a trade secret. Such licensing could include provisions which require them to reveal the use of the idea. (Will they fulfill it?). How much do you expect to be paid?
Re 2. It is worth considering if the output has a visible signature. For example, does the method have a particular form of output, including mistakes, or doe sit need a particular form of input or consumption of resources. It may then be possible to claim these more visible, but less idea-specific items. As an example from beer making (another process which may be kept secret), if you can find a signature of the process in the beer as some small amount of contaminant, then claim beer with with that contaminant.
Re 3. How long will the infringement go on for? The patent can be in force for over 15 years and one can sue 6 years back. Is it reasonable to expect that in all that time there will be no whistle blowers or inadvertent admission or perhaps a worker who left the agency or perhaps a disgruntled employee (all ways of getting hidden information)?
So, to be more specific about strategy. Which, of course, is very fact specific.
(A) See if trade secret protection is an option
(B) try to identify consequences, both upstream and downstream and see if you can claim these (should be done in any case)
(C) If your s/w will not be modified by client, consider artificially generating such consequences.
(D) Try to generate some contractual obligations to provide any type of information.