Anyway, here goes:
The bookshelves lining two entire walls of the study held almost as much interest for him as the rifle cabinet did. Paul or Michelle, or perhaps both, liked to read as much as he did. Uncle Danny had made him leave everything behind, including the book he had been reading. Chris eyed the shelf hungrily. He would have to take a longer look later.
I suppose this isn't a horrible paragraph, but it's kind of a dead one. There are two sentences in a row that end with the "as much as ___ did" structure. It also suggests that Chris is planning on borrowing a book from Paul and Michelle. Now he will eventually do just that, but at this point he shouldn't be planning on it. He should be expecting to be there a day at most. I let my knowledge of what was coming seep into the moment. Also, when it comes to third person limited, it's not particularly tight penetration.
The thing is, though, having decided I didn't care for this paragraph, I couldn't seem to fix it. At first I was mostly trying to find a way to take out the repeated "as much as ___ did"s, so I was just trying to come up with different words to say the same thing. The problem was that I could not say the same things without some of the problems I just pointed out, but had not yet articulated to myself. I didn't know why the paragraph wasn't working, but it continued to not work no matter what words I plugged into the existing sentences.
Then I decided perhaps I shouldn't be trying so much to keep the existing sentences with just a few tweaks. Perhaps what I needed to do was delete the entire paragraph and rewrite it from scratch. I tried that, and managed to get something that I could live with.
Here's the new version. Hopefully it's better:
Though he was supposed to be focused on the rifle cabinet, something else caught his eye. All those books! Somebody here was a reader. It must be nice to be able to keep books after you read them and look at them again later if you wanted to. Chris had never had the chance to take a close look at somebody’s book collection; what you read probably said something about you. Hopefully he’d get to spend some time in this room before he left.