In this context, "made out" normally means "implied", that is to suggest something without explicitly stating it, although as it can also mean a form of deception it could include lying or making a false statement.
A simple way to test which is correct is to substitute the expression for word "implied" (or "stated", either work, but in my examples I will use implied):
He implied like he'd read the book, whereas he hadn't. INCORRECT
He implied that he'd read the book, whereas he hadn't. CORRECT
He implied he'd read the book, whereas he hadn't. CORRECT
Is "made out like..." sometimes used? Yes, idiomatically, and generally in US English, although some British English speakers are influenced by US culture and so it is not entirely unheard of in British English.
"Like" is a comparative term used to say that two things are alike either in one or several comparable ways, but not usually to suggest that two things are identical or the same in every way. If the purpose of a statement like yours is to say that someone suggested, hinted, implied or directly stated that someone is something or did something, then saying their speech made out "like" something brings in doubt. In some cases, that might be what you want to say, for example:
He made out like I was an idiot or something.
I would take from such a statement that the speaker felt they had been unfairly maligned. There is no suggestion that the other party actually called them an "idiot", but perhaps what they said made them look foolish. This is an example of a fairly common expression and is less direct that your examples in which there cannot be any hyperbole or comparison - either someone read a book, or they did not.