I was trying to figure out something else first :-)
Ok, so unit sqlite3conn uses (global) variable SQLiteLibraryName to locate/name the (in case for windows) dll.
However, the use of that variable is deprecated, and we should really use (global variable) sqlite3dyn.SQLiteDefaultLibrary (meaning unit sqlite3dyn). By default (again for windows) that uses const Sqlite3Lib to declare the name. For windows that reads: "sqlite3.dll"
So, that is the first order of business.. e.g to make sure the name matches

However, before tampering with that variable directly, it is stated (
https://www.freepascal.org/docs-html/fcl/sqldb/universalconnectors.html) that:
From this mechanism it follows that before a particular connection type can be used, its definition must be present in the list of connector types. This means that the unit of the connection type (ibconnection, pqconnection etc.) must be included in the uses clause of the program file: if it is not included, the connection type will not be registered, and it will not be available for use in the universal connector.
afaik, that part I have covered in my comments/instructions, e.g. that you should add unit "sqlite3conn" to your uses clause.
then it reads:
The universal connector only exposes the properties common to all connection types (the ones in TSQLConnection). It does not expose properties for all the properties available in specific TSQLConnection descendents. This means that if connection-specific options must be used, they must be included in the Params property of the universal connector in the form Name=Value. When the actual connection instance is created, the connection-specific properties will be set from the specified parameters.
... and I seem unable to locate any evidence of the existence of any code that provides the name of the library using the parameters property

Now, to come back to your question
I Wonder. I'm using 32bit SQlite for sure. My IDE I think May be 64Bit: How can I confirm that.
Lazarus/help/about or Lazarus/view/IDE internals/about FPC (don't forget about the second tab)
But, nothing of that matters. You can have a 16, 32, 64 , 128, or 256 bit version of Lazarus installed

but it will produce a 32-bit application as long as you selected that in your project options.
Lazarus/project/project options, then from the left list select Config and target, and you can see the options at the right: Target OS, Target CPU, Target processor.
That should read win32, i386 and default respectively.
Don't forget to save your changes before (re)compiling.
edit: and there is one other possibility is able to cause failure, even though you use the correct bitness of the dll, namely the dependencies for that dll. see also
https://system.data.sqlite.org/index.html/doc/trunk/www/faq.wiki#q11 :
(11) Why do I get a DllNotFoundException (for "sqlite3.dll" or "SQLite.Interop.dll") when trying to run my application?
Either the named dynamic link library (DLL) cannot be located or it cannot be loaded due to missing dependencies. Make sure the named dynamic link library is located in the application directory or a directory along the system PATH and try again. Also, be sure the necessary Visual C++ runtime redistributable has been installed unless you are using a dynamic link library that was built statically linked to it.
And also with regards to the dependencies that means that those redistributable should match the bitness of your produced executable.