To answer the question as asked, I have the thoughts of the POV character in italics, no matter who's speaking or whatever else is going on. One time I had a character tailing a pair of non-perspective characters in conversation. The character can't see them as they're talking and has to focus on moving around silently, so it's very much limited to his perspective, as the only visual window into the story is his eyes. It was not a drawn-out conversation by any means, but avoiding an overlong conversation, whether eavesdropped or not, is a good rule in general. I suppose I should put this in an author's note somewhere, but thoughts in my stories are assumed to take place instantly, so the insertion of the perspective character's thoughts, to the tune of- That explains why you were lying earlier- in response to a revelation from a talking character does not lengthen the time that passes in the scene. Having the thoughts of the POV inserted into the dialogue as it's taking place seems to remind me at least that we have access to the thoughts of that character and only that character, which is not the case with omniscient narrators.
I'm confused as to why you find your conclusion unsatisfying. Did you wish to write a really long conversation to be overheard by the perspective?

 

You could make the overheard conversation only partly understandable. That way the POV character gets periodic need to interpolate what was heard with some evaluation of meaning; it's like a short mystery puzzle. The scene involves incomplete information, so both POV character and reader must complete it. The trick is finding key pieces of the puzzle to remove that wouldn't invalidate the overheard conversation.

A version I used, however brief:

 

Stuff I'd put in, interspersed with overhead dialogue:

1. moving closer to repositioning to hear more-- "Harry snuck closer, taking care that his sneakers didn't scuff and give away his position"

2. internal dialogue on what the sneaking character thinks of what he's overhearing-- "So that's why they've been avoiding me, he thought furiously. Ron and Hermione are off buggering away while I'm stuck with nothing but Fred and George's toy wands! "

3. hard to do well, but sometimes fragments of dialogue can work:

"Hagrid said ---dog-- don't know where" Ron said, and Harry struggled to hear the rest. A dog? Why would Hagrid care about a non-magical, non-scary dog? He already has Fang.

Fragments of dialogue can get annoying to type out and read sometimes, but I've seen it done well occasionally.

 

I feel that the problem can be solved by calling attention to the inherent change of focus that comes along with eavesdropping. If you were, for example, to hide in a closet and eavesdrop on a conversation: where is your focus? From what I can imagine, you are only conscious of two main things: the conversation you are listening in on, and staying hidden. Calling the reader's attention to those two things would keep them in the head-space of the narrator, while still allowing the author to write a long conversation that the narrator is not a participant of.

I would use language such as, "Harry heard him sigh, before continuing, "I'm not even sure whether or not..."." or "Harry stifled a gasp at that; he couldn't be discovered before he found out exactly when they would be meeting next." or " The thin gaps in the closet door allowed Harry to see the disgust on his face when he said, "Well, I guess we'll have to."."

Hopefully that helps a little!

 

I think you hit the nail on the head when you mention trying to work the information into the story in other ways. Long dialogue isn't bad in of itself, but it has to serve a purpose. If we can increase the exposition without boring the reader, then that makes the most sense.

I really like @wordhammer's example of having bits and pieces of the conversation filter to the narrator's perspective. If I'm listening in on a conversation, it makes the most sense that I might space out for a moment or lose the trail of speech if my attention is pulled by someone else.

 

Both @Garden and @wordhammer nailed it I think.

If the main character is eavesdropping and trying to hear as much as possible (think every Assassin's Creed game ever), it should be long, drawn-out, but also with details like, as @Garden put it:

Sneakers scuffling, avoiding twigs on the ground, maybe doing something that either makes the mission of eavesdropping fail or almost fail.

Indeed, the observer's thoughts should be interspersed among the other bits of descriptive detail. And, of course, the view and thoughts the eavesdropper has should be biased, unless it isn't for some reason.