You can encrypt and control SQLite access. It's not ready, you have to do this process, but it's doable.
I would think twice before discarding it. Other options may have more facilities, but if you do not know what you are doing you will not have security. Remembering that if the person has access to the file and there is a password known to people or even inserted into the machine in some way, security will never be 100%.
I would not use MDB, but you can use SQL Server Compact (which is already legacy but still supported), or Firebird likes it. There is a version to work on board. It has limitations, but I think it will suit you.
LocalDB is an option that will work well and is best suited for those who are familiar with SQL Server. Of course he's a lot heavier. It leaves a bit of what we call the embedded database. It is not characterized by being a desktop database. But only you can tell if it is suitable for your situation. Obviously it has limitations of SQL Server Express, but it caters well to almost all desktop situations if it accepts the weight of it.