Is there a Solid State amp that can satisfy a SET guy?

How much power do you want?  A parallel 300B will get you a bit more power...20w.  A parrallel 520B using EMLs will get you to 28w but these are not cheap.  New, mine are $23K in Stereo but we usually have a used Argento or two floating around that can be had in low teens.  Just a little above your budget.  I would just need to have the power supply swapped for US.  

Right now I have this listed for a customer which is an interesting amp:

https://www.audiogon.com/listings/lis9j5a1-art-audio-adagio-stereo-set-amp-27w-tube

There were only a couple made and it probably the most power you are going to get from a 300B family tube. In the end, I get that you want to get to about 50w of power which I can't do in anything other than a Push Pull design.  

BTW...I understand what you mean about 845 monos.  I run 45w 845s that are push pull and it is definitely more linear than a 300B.  The 845s are going to be about where your warmest SS amps are going to fall.  If you don't like 845 monos I am not sure you will ever be satisfied with a SS amp.  

I would like to draw your attention to the german AVM Audio brand.
I run an Ovation A6.2 into Dynaudio Sapphires. The craftsmanship of these amps is outstanding. Their class A/AB is amps are very revealing yet very musical. The top of the range amps, AVM Ovation line, combine SState and Tube ouput. These are state of the art/benchmark products that last a lifetime. Very special.

Hope you have the opportunity to auditio.

You can try Pass Labs. Some rolled off highs and bloated bass, just like a SET amp...

austinstereo
As a result of being in sales and service of audio gear for a very long time, we have yet to encounter a solid state amp that can match a tube amp in terms of warmth and space. Just as true, we have yet to encounter a tube amp that has the tautness and attack of a solid state amp. A push pull tube unit is a bit more of a happy medium between the two extremes.

>>>>>You might consider widening your search as a good tube amp will meet your requirements for tautness and attack, obviously the usual audiophile protocols must be followed, you know, room treatments, EMI/RFI treatments, vibration isolation, etc. as a large part of the sound ultimately is a function of “system engineering.” “Plug and play” only gets you so far. 

Listen to a zero-feedback design. Close, if not quite matching, SET sound but also with advantages in bass control. Ayre and Aesthetix are in this category (altho the Aesthetix is a hybrid).