From B&W 804 D3 to Focal Sopra 2?

The anthem is fine and enough power for either. You are using 1-5 watts for most listening. More power will not improve the bass. 
 

Have you tried the room correction in the STR preamp? That has the potential to increase the bass. The STR also has sub management and I personally would just add subs and highpass them in the preamp, this will take the system to another level. 
 

As for the 804d3 I find them lean in the bass and bright on the top end. The Sopra is a much smoother and all around better speaker. How ever the Sopra 2 is not a bass monster by any means and I also find it a bit weak without subs. But it will have more bass than the 804 and be much more balanced. 

For the same money you could get the Revel 328be which is much larger and of the same caliber in sound as the Sopra, the 328be will not be lacking for bass. They are often on sale for 30% off if you look around. 

My overall impressions are mixed.

I did an in-depth analysis on Focal’s years ago, and have been watching the reviews and measurements ever since. My impression is that Focal deliberately makes their speakers either "hard to drive" or "scale with amplifier." Two terms that mean the same thing.

Both the B&W and the Focals also suffer from odd off-axis oddities which limits the sweet spot.

If you want the best out of any Focal speaker you should be prepared to get a sturdy amplifier. I’m not sure how the Anthem will fare honestly, it may be fine. I am with James though that you should consider Revel and Monitor Audio as excellent counter points that are easier to drive and will give you a wider listening window. They are also going to sound somewhat laid back at first.

Otherwise, I think that the B&W and Focals each have their quirks and I don't see switching these speakers as much more than a lateral move.  IMHO, keep what you have for now and switch to something your ears tell you is a real positive improvement.

Thanks James, Erik. Good food for thought. I don’t think I’ll get those Focals after all.

Sounds like you’re looking at the Focals out of convenience/availability more than what they can do for you sonically, and there are many other excellent options out there.  What’s your budget and are you looking for new or used?  Alternatively, if you otherwise like the 804s you could add a couple good subs and that’d solve your bass issue. 

I don't know what B&W models you auditioned, but I have never heard them described as weak in the bass! I am speaking solely of the Diamond series, though.