Rega Aetheos, Simaudio Moon 340i or NAD M33 for B&W 805s?

None of those three options is particularly bright.  Tonally, they are exactly what you are looking for.  I am not sure that Rega is the best option because I have read it struggles a hair with lower impedance speakers but I do not have direct experience with that.  That would be my only caution.  I am not recalling where I read it but for some reason I am thinking that sub 6 ohm speakers are not ideal. 

There should be no matching issue with SimAudio or the M33.  

The other option you could look at it is Naim.  Both the Supernait and the Uniti series are shallow and tonally spot on what you are looking for.  The latter of course has built in streaming capabilities.  

I like Naim because it thrives with low impedances due to its matching with Focal.  

Full disclosure, I am an NAD and Naim dealer.  I have no relationship with Rega and Simaudio other than I am a happy former owner of a Brio R.  

It was not from a review, it is in the manufacturers stated specifications.  

Power outputs at rated 230/115 V supply voltage:
125 W RMS both channels driven into the rated load of 8 Ω
156 W RMS both channels driven into 6 Ω
Note: Continued high level use into 6 Ω may cause the case to exceed 40 °C above the ambient temperature and activate the thermal shutdown.

The min impedance on the 805 D3 is 4.6 ohms so not particularly low but low for pairing with this amp, problematic.  

@capt57 I would look seriously at the Supernait or a Uniti Star or Nova with a major caveat that they are out-of-stock.  The absolute power rating isn't that high but it is plenty of power to drive those 805s and tonally is warmer and a brilliant match.  The only issue is that they are out-of-stock at the distributor.  The Uniti Star looks like inventory may be in soonish.  The Supernait has no estimated delivery date. 

An alternative at $6299, the AVM CS 2.3 is an all-in-one streaming unit that would definitely do the job.  AVM is tonally quite warm and a brilliant match for B&W.  This new unit has been worked on a long time to make sure that it is virtually bug free.  My guys at the distributor have been running it for a few months without issue and have been super happy with the software.  Silver and Black are in-stock and it would be a nice option for you as it has similar features to the M33 or the Uniti Star. 

If I can think of something else I will mention it here.  

@patrickdowns The Mu-So is in-stock at the distributor in three colors.  It is a nice unit.  I feel it is MUCH better than a Sonos or a Pulse but has the natural limitations of a single box.  If you can live with that, it is a good sounding option.  

@capt57 Topline (8 ohm) output means very little if the speaker is lower impedance.  It means more current is required and its rating and ability to handle lower impedance is far more important.  In the case of the Aethos, it needs very easy to drive speakers or it is going to over-heat according to the manufacturer.  

The Supernait is rated to 2 ohms and is designed to be paired with Focal.  They are relatively difficult speakers to drive with impedances that will bottom out below 3 ohms.  The 805 D3s will be an easy load on them.  

The difference between 80w and 125w in terms of peak output is about 2 dB.  At 80w, the B&W will output 103.4 dB at 10' and at 125w, the peak output will be 105.3 dB.  In the extremes it will make a small difference.  

Musical Fidelity is wonderful gear but slightly forward in terms of the sound profile.  It is not a brand I would match with B&W.  The other issue is that it is very difficult to find lately.  

I was thinking that McIntosh almost assuredly has an integrated in this price range but it might be too large.  

Are you willing to consider a hybrid class D?  Tube input stage with class D output stage.  I have an amp that is exactly 16" deep and it is 100w into 8 ohms, 150 into 4 ohms.  The Canor AI 2.10,  I also have a demo of the AVM Evolution A5.2 integrated which I think is 185w into 8 ohms.  It is 16" deep including nobs and binding posts.    I have used both to drive Vivid Kaya 45s and they have a min impedance of 2.8 ohms so much tougher than the B&Ws. The Canor is $4K.  The AVM is my demo.  

I have not hear the Supernait with the 805 D3.  I can't answer how that would work out.  I am making a recco based strictly on a paper matchup.  

I found the depth of the AVM, it is 325mm or approximately 12.8".  I know if has redundancy with the CD player and it is a bit over budget at $6300 but it does include a good DAC and has built in streaming.  

I am largely out of ideas beyond that.