Based on those assumptions do you have any problem with a local moving the box off NPS property and communicating with you on clue changes?

Why? Where is there an indication that LBs are not allowed? As has been pointed out (WITH A LINK), NPS no longer explicitly prohibits them, and it is up to each park to make their own determination. Why remove someone else's box based on "well, maybe someone somewhere has an unwritten and unposted prohibition for this specific park, based on word of mouth"? How does it "look bad" or generate "ill-will", if the rules aren't posted and published?

You have been made aware of the issue by the local planter.

And therein lies the conundrum.

The local boxer is "certain" of this policy prohibiting letterboxes yet there is not "official" mention of the policy on any park related listing of policies related to the prohibition of which they speak.

To me, if it isn't in verifiable format, it isn't actually a policy.
::shrug::