Well here is what I do this in a few of my projects - something that has proven reliable enough for me over the years (which you may or may want to do as well):
I have the program itself create the database files in an initialization routine. First however, it creates the sub folders in which the database files will be stored, if they don't already exist.
To do this, the program just checks if the folder exists and if the database file exists and if they do not, it creates them on the spot:
If Directory.Exists(gSQLDatabasePathName) Then
Else
Directory.CreateDirectory(gSQLDatabasePathName)
End If
If File.Exists(gSQLiteFullDatabaseName) Then
Else
...
I also have the program do some other stuff in the initialization routine, like creating an encryption key to be used when storing / retrieving the data - but that may be more than you need (also, for full disclosure, this has some rare issues that I haven't been able to pin down).
Here too are some addition considerations:
I appreciate you have said that you want to give the user the choice of where to store their database files. However, I would suggest storing them in the standard locations
Where is the correct place to store my application specific data?
and only allowing the users to move them if the really need to (for example if the database needs to be shared over the network) as it will make the support of your app harder if every user has their data stored in different places.
I have found letting the user see in their options/settings windows where their database is stored is a good idea.
Also to encourage them to back those files /directories up.
Also to create automatic backups of several generations for the user.
Hope this helps.