I've found how to solve that looking at AdminTagProcessor::CheckPermCache method, which decides whatever we need to rebuild category permission cache when catalog is visited.
We simply need to put ForcePermCacheUpdate mark into system cache (setDBCache method) and then once "Structure & Data" section will be visited + this mark present automatic cache rebuild (with progress bar) will start.
This way if we can do silent cache rebuild we do it, but if we can't we just put a mark to indicate that it needs to be done.