I think .dll bitness should match that of operating system --- Windows, I assume.
I put 64-bit sqlite3.dll under c:\windows\system32 if it is 64 bit. (32-bit one should be under c:\windows\SYSWOW64. I'm not confusing), Because these folders are in the search paths by default (if not absolutely default, they are almost).
Sometimes I make copies of exe file to USB or CD. In those cases, I copy sqlite3.dll to the same folder as exe file. But on my development PC, I do not put sqlite3.dll in the Lazarus directory, project file directory, or exe file directory. The exe file may run on other directory than that the exe file resides.
This is my practice. You may try other approaches.
And SQLite seems single-user system. If it is opened by dbbrowser, etc., you cannot open it in Lazarus (based on my experience which is limited). So please check these points first.