I'm not a balisong person, and it is unlikely I ever will be. Nevertheless, IMO, the biggest problem in most of these anecdotes, at least where the LEOs involved asked about weapons, the answer should always be, "NO."
Your knives are tools. Should the situation ever require self defense, then hopefully the knife/knives you are carrying possess some ability to serve as weapons. But, until that point, they are not weapons. So, truthfully say, "NO."
I used to watch the TV show COPS frequently. I was always amazed by the almost universal affirmative response when people are asked if the officer(s) may search the car. One show was on state troopers. They were on the lookout for cars which appeared to have high mileage, driven by non-white individuals. This was the sole basis for "probable cause." They checked the odometer to verify the high mileage, then asked to search. They obviously knew what they were doing, because invariably the car was muling drugs. But, if the driver had said, "No" to the search request, nothing would have come of the stop. Of course, with the TV camera rolling, the officers weren't about to do anything illegal. I can't help but wonder though, about those cars stopped when there was no journalist present.
A year or so ago, an expose was done on both the Louisiana State Police and small town gendarmes on a major highway crossing several counties. An extraoardinarily high percentage of the stops and searches, etc, were flat illegal. Of course, that state has always had a problem with corrupt police.
One of my friends is a retired LEO. He works as an investigator. He knows the laws better than most active LEOs. Frequently when he is performing surveillance, in a jurisdiction where the police are not highly thought of, or it is known that one or more members of the force are related to the subject, he not only says, "No." He says, "Hell no! and get out of my face." He has many friends among LEOs, and a few enemies. But nobody has had the hair to take him in.