it currently is only possible to restrict access to a database by removing the write-right on the filesystem for the db-file. So under linux, chmod it in a way so that the webserver-user cannot write the file, on windows, most easily make the file read-only.
But this also means the admin user cannot edit the db any longer, You can have a read-only copy, but then you need to replace the copy each time you edited the original file.
If this is a showstopper for you, probably it is easier for you to implement rights management in phpliteAdmin instead of starting from scratch. Of course if you find an alternative solution that works for you, that's also an option and I'd be interested in this as it is always interesting to learn from what else is on the market. I don't know any script that is comparably to phpLiteAdmin and has this kind of rights management, but this surely does necessarily not mean there is no such script.
Although I am part of the phpLiteAdmin project, I won't keep it secret which other similar scripts I know. The most important one in my opinion is Adminer. It supports rights management for dbs that have their own rights management like mySQL, but I am unsure whether they have rights management for SQLite. But maybe it's worth to have a look at it.
If you implement rights management for phpLiteAdmin, of course we would be interested in the implementation. If it's generic enough for widespread use it has good chances to make it into the next version.