Home Theater Receiver recommendations for B&W 803S?

Agree.  That Denon AVR-4308CI is already at the top level of receiver power supply capability.  You are not going to get anything better by upgrading to something newer, other than support for 4K.
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It would be better to get a really high current multi-channel amplifier for those B&W speakers.   If you are looking at new within $2k, the best I could recommend is the Emotiva XPA-DR3 amp.  It's a fully balanced / differential 3 channel amp.  Then use the receiver to power any surrounds that you have.  There are others and if you look at used, then things get complicated as all the amps sound radically different.  Emotiva is obviously not the best amp, but it is extremely good for the money and very neutral sounding (not too warm, not too bright) and has a lot of current and attack/detail.

Thanks @auxinput!

I just came across a listing here for a Bryston SP3... which looks like it could be perfect, actually (if I'm willing to fudge the budget a bit...)

https://www.audiogon.com/listings/lis96275-bryston-sp3-surround-processor-with-4k-sdr-hdmi-switching...

Curious to know what your thoughts are on that one?  

I actually tried the SP3 in my system.  Before I speak, you should probably know that I am extremely highly critical of equipment and sound - to the point where others say that I'm splitting hairs, lol.  A sound extremest, if you will.

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That being said, I tried working with the SP3 a lot.  It definitely had very good sound quality, probably the best bass I ever heard from a processor, but the sound was just too laid back for me.  I could hear that the Class A audio stages were very good, but I just did not get enough attack / slam / detail in the mids and upper mids for my tastes.  Also, I don't think I was getting good resolution output.  I believe the Digital/DAC board is compromised in the SP3 because they use an off-the-shelf digital board from Momentum Technologies.  So, I think the sound will not be quite as clean as other processors.  With all that, it could be that you would like the SP3 just fine.  It has a big following over on Audio Circle and AudioAficianado forums and many people rave about the SP3.
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Ultimately, my older Krell HTS 7.1 was highly superior in sound quality, but was very old.  After trying multiple processors, I run with a Krell S1200U, which is just about the best sound quality processor in my opinion.  However, it is just a little older and doesn't support 4K.
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Because of your 4K requirement, it is going to be difficult to find a good processor in your budget.  I think the only thing I would recommend is to go get a previous generation Marantz AV8802a.  They are readily available "new-in-box" from ebay and sellers such as Crutchfield at the $2500 level.  They are excellent sound quality, has very large main power supply and excellent local power supply capacitors on the DAC I/V board, and also has fully discrete analog output stages.  The Marantz has excellent sound quality, but is voiced to be warm sounding.  The upgrade the fuse to something like Hi-Fi Tuning Supreme or Silverstar to reduce the "warmth" and increase the clarity/detail.  I think that's about the best you can do.
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I tested the latest Marantz AV8805 in my system and was quite impressed.  It had very good slam and impact, but my Krell S1200U just beat it on high frequency clarity and detail (something that can be easily fixed by fuse upgrade).

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Like I said before, probably the first thing you can do is to buy a multi-channel amp and connect it to your Denon receiver using the Denon "pre-outs".  That would give you a big jump in ability to let your B&W speakers shine!  Then you can always replace the Denon with an HT Processor in the future (with additional amps for the remaining channels if you need them).  Many amps will have a 12V remote turn on trigger, so the only thing you need to do is click the "power" button on the HT Processor remote and all your amps will turn on automatically (so it will almost be like using a receiver).
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Or you can get HT Processor and amp(s) at the same time.

Oh one more option that caphill likes to recommend is the Classe Sigma SSP processor.  It supports 4K, however only supports HDCP 2.2 through one input (HDMI input 4).  It is also a very nice processor, but as good as it is, it does not have discrete analog stages like the Marantz.  The Classe is quite a bit smaller, so if size is an issue, than Marantz may not be an option.  The Marantz is definitely a beast.  The Classe Sigma SSP is in the same $2500 price range.  There's actually one for sale from TMR Audio for $2408.

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In a comparison, Marantz AV8802a with fuse upgrade VS Classe Sigma SSP, I honestly do not know which one would win.  Personally, I would actually go with the Marantz and work with fuse/power cord upgrades because of the fully discrete analog stages.  In all my R&D and testing with circuits, as good as monolothic op amps can be, they never reached the sound quality of fully discrete analog stages.  My own personal opinion, of course.

@auxinput @awilder

The current model Classe Sigma SSP Mk2 retails for $6k new and even used it will be in around $4k ish since it supports the latest formats and is current model as far as I know. I’m worried that the Sigma SSP that you saw for sale for $2408 was possibly the original first gen Sigma SSP that does not support 4K and all other current formats, that’s why it’s being offered for sale for $2408 because it’s been replaced by the Sigma SSP Mk2. The original retail price of the Sigma SSP (first gen) was $5k.

The Classe Sigma SSP is optimized for stereo. If the OP focus more on home theater I would go with the Marantz AV8802 as it’s also easier to calibrate for surrounds using its Audyssey auto room correction. The Classe lacks auto room correction but provides manual PEQ instead which requires proper tools and equipments and knowledge on how to implement it. The Marantz AV8802 is much easier to calibrate using its built-in Audyssey even though I’m not a fan of Audyssey at all.

For stereo music, we did compare the Classe Sigma SSP with the Marantz AV8802 and both processors were paired with Classe Delta CAM-300 monoblock amps (class AB) on B&W 802 D3 speakers before at my friend’s shop and the Sigma SSP sounded better for stereo music playbacks more musical and better sound overall. We used the Sigma SSP’s USB input to stream music and we did the same with the Marantz AV8802. The Classe Sigma SSP also has better DAC configuration/implementation at least for the front LR channels and has superb USB input implementation in its signal path, which is copied and pasted from Classe CP 800 stereo preamp/DAC. 

But for surround sound movies (home theater) both of them performed great but I would recommend the Marantz AV8802 for the OP if his primary focus is home theater and it’s easier to properly calibrate using its Audyssey auto room correction. The manual PEQ on the Classe is much more difficult to properly calibrate and becomes a deal breaker for some people.