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The story is third person limited to Bob's point of view. He is with Alice, and I have the following line.

Alice sat up proudly, then shrunk back down, realizing where she was.

Compare this to

Alice sat up proudly, then shrunk back down, apparently realizing where she was.

This is a tiny change, but without it I am technically breaking the limited scope and seeing into Alice's head briefly. But at the same time, Bob could realistically realize that was what was going on in Alice's head, so it does not seem egregious.

The reason I am hesitant to do the second is because I do not want to be constantly hedging statements with things like "apparently" or "seemed to" since it can bog down the writing if used often.

So the question: Is it okay to show thoughts of the non-perspective character if the perspective character can reasonably guess them?

You seem to have chosen third person limited scope, and then decided that you're "supposed to" follow the rules of third person limited scope. You should make decisions based on what serves the story, not based on whether it's consistent with a label that otherwise applies to your story. If you had some motivation for choosing third person limited, then ask whether putting in "apparently" serves that reason you originally had for third person limited, rather than asking whether it is needed to qualify as third person limited. If you think that distancing the reader from other people serves the story, then put in qualifying words such as "apparently". But if you think the story is better served by being more direct, then be direct.

Also, another option that is somewhat risky as it can erode reader trust, but if done well cements Bob's point of view even more than distancing words such as "apparently", is to give Bob an unreliable POV. If you give narration from Bob's POV, and don't explicitly qualify it as being from Bob's POV, and the narration is later shown to have been occasionally inaccurate, the reader will pick up on the fact that when narration strays into other people's motivations, it has to be taken with a grain of salt. This puts the reader more into Bob's POV, because Bob isn't constantly qualifying his evaluations of other people's mental states. He's not thinking "It appears that Alice realized where she is", he's just thinking "Alice realized where she is". He's making assumptions about what people are thinking, and trying to create a model of their mental states, and occasionally realizing that he was wrong.

This will be a matter of opinion. My opinion is no, it is not okay. I write in 3PL myself, exclusively, and everything I write is as if the MC is seeing it.

You are doing nothing but saving space, and saving space is not important. I would probably write that scene as:

Alice sat up proudly, then shrunk back down.

She just realized where she was, Bob thought.

But, since Bob is the only one that can think [whose thoughts are ever shown], and this is indicated by italics, the following would suffice:

Alice sat up proudly, then shrunk back down.

She just realized where she was.

Then, because things should be filtered through Bob's mind, I'd add some take on it; Bob should feel something about what he has seen. Not just report it. Perhaps he's sympathetic.

Alice sat up proudly, then shrunk back down.

She just realized where she was. Too bad, anywhere else and she's got a right to be proud.

My narrator is not the MC, and the narrator can describe visual things in ways Bob would not. For example, "sitting up proudly". I can't be certain Alice is "proud", but I can be just as certain, visually, as saying Alice is Angry, afraid, grieving, etc. So I don't mind the attribution of "proudly".

I also don't mind if the 3P narrator gets poetic or imaginative in their descriptions of a setting, in ways the MC would not.

But as far as knowing what is in any character's mind: that is what the "LIMITED" means. Find a way for Bob to interpret what is in Alice's mind. And make it clear, don't make us infer what Bob inferred.

Welcome!

+1 to both existing answers.

The advantage of adding the word 'apparently' is that it further anchors us into Bob's mind without the need to add italicized thought (which can become distracting in its own right.)

But--there are many tricks to get around the point-of-view break. You've mentioned one.

1. Alice sat up proudly, then shrunk back down, apparently realizing where she was.

Some additional possibilities:

2. Alice sat up proudly, then shrunk back down, perhaps realizing where she was.




3. Alice sat up proudly, then shrunk back down, saying, "I just realized where I am."




4. Alice sat up proudly, then shrunk back down. Bob thought to himself, She must recognize this place.




5. Alice sat up proudly, then shrunk back down. It seemed that she recognized the room.




6. Alice sat up proudly, then shrunk back down, wide-eyed.

(And as an aside, you can strike 'up' and 'down' if it is a space concern.)

7. Alice sat proudly, then shrunk back, as though realizing where she was.

The more varied you get in your writing, the less of a distraction it will be. I do personally find small POV breaks distracting and rarely see them in current books. I see them more often in older books, and maybe the conventions were different.

As a reader, I wouldn't think twice about that brief "break" of POV. I also don't generally analyze "what point of view is this in?" while reading. I think, if it works in the story and flows well, don't over think it.

I just recently finished an award winning book that is third person limited, but for about three pages I spent time in a minor character's head. That "broke the rules", I noticed it, but didn't mind because I liked the story. It helped moved the narrative along.