will changing to solid-state make the bass in my wilson sophias better at low volume?

Tubes are great for preamp use, but solid state amps control the bass much better. I have had some very good tube amps and none of them produced quality bass, too sloppy. I would stay away from loudness controls, I’ve had a couple in the past. I had the same problem in a smaller room, low volume, no bass, turn it up for good bass, too loud. Got a couple of rel subs and tuned them for low volumes and they still worked when the volume got louder. Now you can get remote controls for your subs so you can adjust the bass for any volume

Why not install Amarra Luxe, as it works on top of iTunes? It sounds TONS better than iTunes, and also has a number of choices for digital tone control, including various curves optimized for various kinds of music. It also has digital volume control, and allows for over sampling and for MQA full unfolding. 

Always had this problem. The first octaves to cut out as you lower the volume is the low bass. I finally spent a few bucks more and got a much more powerful amp. It is SS and it is rated above the max of my speakers wattage, but now when I lower the volume octave to octave it is still present. No loudness even exists on my pre-amp. I am using 500 watts into 4 ohm nominal speakers, rated at 300 max. Just an observation that is a first, and a pleasant discovery indeed.

Presumably you have optimised speaker placement -- if not, you could be losing some bass energy due to placement.Sophias output a reasonable amount of (upper) bass.
Otherwise, missing bass at low amplitude also means your (all of our) ears are less sensitive to lower-mid and bass frequencies (Fletcher-Munson curve) as mentioned above.

Once I demoed the Sophia 3 with a musical fidelity A5 (solid state integrated amp) back to back with a $30k AR tube pre and amp (model?). The tube setup lost most of the dynamic bass slam that I found intriguing on the A5. The bass had crazy slam and power on the A5 in a very large room. 
Maybe try a cheap use solid state amp, even a class d just to mess with. If you like it sell it and get a good solid state amp. It is alway a compromise somewhere though. Expect highs to suffer.