the alternatives are 2:
- you find a work-around to find that specific string you need inside the sqlite3 db, something like I did for steam and Phrase (good luck with sqlite3 ih ih ih)
- you write a minimalistic sqlite3 db client using less instructions as possible
the alternatives don't seem much interesting, no?
anyway there is a third alternative which could be interesting to you although it's probably not so fast and immediate to implement (so if you are not enough advanced with your programming language, give up).
practically you can store sqlite3.dll directly inside your executable using two possible ways: as a resource (most commonly used, personally I have never used it) or as a fixed buffer (I guess this way has been implemented only in C and, since I tested it, works very well).
so when the executable starts it's only needed to manually link the functions of the dll you want to use with the dll inside your exe (I don't know if when it's included as resource it can be linked directly in memory or, more obviously, must be extracted first).
if the size is a problem is enough to compress the dll or directly the whole exe with an open source packer like UPX:
http://upx.sourceforge.net/