看穿 and 看透 are hardly used in the literal way. They are almost synonymous as I can't think of any instances where they're not interchangeable. 看穿 has a slightly more derogatory tone as it's often used against tricks or lies. 看透 can be used against philosophical subjects such as the meaning of life.
I think the most natural thing to say about an invisible man in Chinese is that he is transparent or invisible: “他的身体是透明的。”
If you want a phrase akin to "seeing through", it should be
(透过[X])看到[something inside/behind X]. It sounds natural but the focus is on the object being seen rather than the transparent medium. When you try to emphasize the medium, this phrase feels like explanation or development for a previous declarative (such as "he is transparent") rather than a standalone one. Also this phrase doesn't necessarily mean [X] is transparent; alternatively [X] can be a fishnet or the viewer has supernatural vision.
Examples (all of them are optimized for natural-ness and fluidity rather than syntactic completeness, but I asure you they're all grammatical correct):
[Advanced tips] There is a widely known phrase
透过现象看本质 (commonly translated into "see through appearance to perceive the essence"; 看 here is more 'find', 'seek' than 'can see'.) which sounds very serious and has a strong textbook tone. If you say
透过[XX]看[YY] where XX and YY are both two-character words, depending on the context it may (unintentionally) either inherit the seriousness, or become sarcastic, or sound like a ragged verse. This can be avoided by simply not using two-character words or replacing 看 with 看到/看见 or add fillers in between of the sentence to break the similarity.