I'd not bother with Klipsch subs unless it was their best models and even then only if discounted well. In the UK I'd look at BK Elec subs.
I'd still run all sources thru the avr, they make better hubs than tvs do, but YMMV. If the avr has all the features you want, at a price you like, go for it....personally I like the features of Denons more (particularly the room eq software compared to Sony's DCAC).
In a bazillion years I'd never doubt the advice of this man. But...
I just want to add, for what it's worth, you won't fail miserably if you decide you want to keep your speakers all the same brand. I have had a few of these subs. They aren't great by any stretch, but they do function and if as you said you're on your first system and not about volume, this sub could be tolerable.
I currently have one setup in my secondary viewing area that had 2 of the Klipsh 10's and they don't chase me off, annoyed and itching to replace them. It's a relatively small room as well though. Having said that if I had known I'd end up with 2 I would have done better to buy the 12 in which is a much better sub. Though again, nobody here but me would suggest.
Most folks here are VERY particular about thier products and recommendations as they should be, and you probably won't find anyone but me to say this. Which is exactly why I do. Just to have all perspectives.
Again, just to be clear, if you want to spend more on a sub, less on your AVR, by all means get a better one. Even a better Klipsch. But if you're not going past 60 - 62 on your AVR volume dial, (which frankly I never do watching TV) then this entry level sub will work as a starting point. (especially if you think you may have a recycling purpose for it if and when you decide to upgrade. Like a stereo setup somewhere (again in a small room, it's only a 10"after all.) that could use a little punch and can take this repourposed sub.). I'm all about second lives for products.
You didn't mention your room size. And because I have a similar setup and all Klipsch as well, I'm thinking only that it's not entirely out of the question.
Anyway... Not disagreeing with lovinthhd ever. But if you dont have a big room, and place it properly, and aren't blasting volume... It ain't kill you. Especially with a rug and or carpet I think.
You're on track with the Denon. I'm less familiar with thier model numbers but if you can find a 2019 model, you'll clean up on a sale price I think. Especially with your $1200 budget.
Last comment. Maybe I missed it but did you list a Height speaker in your post? I saw the Atmos preferences, but don't recall seeing the ceiling bounce speaker listed. Assuming your not cutting your ceiling..?
Not sure how much traction you'll get on this post, but I can share pics or certain other info if you want on my similar setup to what you're trying for.