I am aware you mentioned bookshelf placement as a requirement and for that I don’t have a perfect recommendation. But I can share my very recent experience trying to find high-sensitivity speakers with a small foot print that would work well with my very low power Decware Zen Triode (2w only) in a substantially smaller room than yours (11x13).
I did purchase the Klipsch RP-600M but placed on stands. While I did enjoy the sound, I was immediately taken aback when doing some AB testing with my other speakers which have a rated sensitivity of 88dB, realizing that Klipsch’s claim of 96dB sensitivity is a total exaggeration. I don’t have the means to do measurements, but it was blatantly obvious that their sensitivity is probably 89-90 dB. They played BARELY louder than my other speakers and I don’t understand how Klipsch is getting away with this, still until this day advertising them as 96dB speakers.
Important for you to consider is the port situation. If you really end up putting your speakers inside a bookshelf, you should pay close attention in your speaker shopping and probably seek front-ported or completely sealed designs over back ported, not one like the Klipsch with it’s huge Tractrix port in the rear. Most speakers will sound better pulled away from the wall regardless of their design, but I think it’s safe to say models that should definitely not be butted against the wall (if you want to utilize their lower frequency output) are designs with a port in the rear.
In my case, I returned the Klipsch and after much online research and consideration of the usual suspects Tekton, Zu, Omen etc, I found a fairly obscure company called NSMT and took a chance buying one of their least expensive, very slim floor standing speakers called The Model 25/Chorus as B-stock for $850 (incl shipping & outrigger feet). For their modest price I am quite happy with them; they are a two-way design with 8” paper woofer and soft dome tweeter and claim to have an extraordinary 98dB sensitivity - which again I can’t prove or measure, but they clearly perform much better with my little Zen Triode amp than the Klipsch did and image really well and have an amazing form factor for my small room.
https://nsmt-loudspeakers.com/The-Chorus
I do think though that they may not provide you with enough bass, depending what sound you are looking for and the music you listen to.
Another idea, to stay within your $800 budget, would be getting a pair of the “unbelievable for the money” Sony SS-CS5 speakers (beware rear port as well, which you would possibly need to plug if placed inside a shelf) that you can grab for as little as $100 and spend the rest on a nice sub, like a REL T/5x ($680). This may sound like a weird idea, but I own the Sony’s and they are quite spectacular in terms of tone and resolution and paired with a good musical sub, you could probably have a rather impactful “audiophile-ish” room filling sound without any visible large boxes. The advantage of this route is that the REL is top notch and could remain in your system for a long time, even if you eventually upgrade other components. If you enjoy youtube reviews, this one is a fun watch of a couple reviewers comparing and riffing about the Sonys. https://youtu.be/ytsCGxfoKHc?t=84