bicycle=yes means: cycling allowed. bicycle=designated means: the path is a cycleway. If a cycleway passes through a special area with an extra usage condition, you just add the appropriate access tag. It will still be a cycleway, still designated, and besides that a usage restriction narrows down the access.
This can only be a problem where a highway=path is used for a designated combined way for cycling and walking. If that happens a lot, in a country where this is commonly done this way, I guess the mappers will have found a solution by now.
In Nederland, it is a rare exception, and will be handled by tagging the most restrictive access value for bicycle and foot. This will still convey the information that it is a path, defaulting to bicycle=yes and foot=yes, and add the other restriction (permissive, destination, customs). You then lose the designated information but keep the access. In the few cases I have seen it’s about a separate path, not accompanying a road.
I don’t know how often this happens in other countries. So far, not many cases have been presented.