Finding a language that is $NP^L$-complete
What you are looking for is a relativization of the Cook-Levin theorem in the sense that you consider SAT and NP relative to an oracle for $L$. You can find the answer to that in this answer, point number 2. There you get (an extension of) Circuit-SAT as a complete language; reducing this to your version of SAT should not be too hard.

Note that, as mentioned in the linked answer, this is a bit different than a relativization in the sense that it is the reduction which possesses an oracle for $L$.