To me, “underlying” is the translation of “sous-jacente”, while “basic” and “fundamental” indeed correspond to “basique” (or rather “de base”), and “fondamentale”.

I think that there is a slight difference in meaning, in that “sous-jacente” stresses the fact that the theory is the root of what you are talking about, and that this link is not so obvious (it lies underneath).

On the other hand, “de base” and “fondamentale” are more neutral and can be used in an absolute way, without relating to something else.

You can say “C’est la théorie fondamentale de M. Lambert”, but you must say “C’est la théorie sous-jacente du discours de M. Lambert” (for example). Or at least the link must be implicit if not expressed.