There are two issues here: 1) whether you would eliminate the "mushy" sound by increasing the amount of warm-up of your solid-state gear, and 2) whether your BAT amp has enough power to properly drive your Dynaudios.
Starting with the second question first, wattage ratings for equipment are usually misleading and often completely irrelevant. What matters most, in the case of a solid-state amp, is the amp's peak power capabilities, which is determined by the size of its power supplies, and in the case of tube amps, the size of its power supplies and quality of its output transformers. There generally aren't surprises - the really expensive gear is where you find amps rated at 100 watt/channel that are subjectively more powerful than typical 400 watt/channel amps. BAT makes pretty fine gear and I suspect that you are not experiencing a power gap - Dynaudios do like power, yes, and are only of average efficiency, but they also have first-order crossovers and don't present a vicious load. And all things being equal, there is very little difference between a 395 watt/channel amp and one that's 300/watts/channel - if you double wattage, you only get 3 extra decibels of volume.
As for "mushiness", standby functions on solid-state amps tend to keep the input circuitry powered up, but not the output circuitry - unfortunately, it is the output transistors that raise the temperature of the amp, so if you toggle in and out of standby, it's the equivalent of turning the amp on and off, with the resultant harmful thermal cycles (heating up and cooling down). Over time, numerous thermal cycles will make internal components more likely to fail. The other problem, of course, is that equipment which has not yet come up to temperature doesn't sound very good, especially in high-resolution systems, and reaching thermal stability can unfortunately take anywhere from four hours to a week, depending upon the piece of gear in question. These are just general observations, however: I do not know how the standby function works on the BAT amp; there are some solid-state designs, such as Class-A biased amps, that are better turned on and off because they run components near their thermal maximum, which increases the likelihood of failure; 24/7 operation will decrease capacitor life (although caps are relatively cheap and easy to replace, and far less worrisome than losing an output transistor). I have contributed to numerous threads addressing whether or not to leave equipment powered up 24/7, and I suggest you look at my "Threads Answered", at "Amps/Preamps", if you want a lot more detail. The short summary of my views is, in the interests of sound quality and in preventing premature component or tube failure, as applicable, I advise leaving most solid-state gear, and most tube gear that uses only small-signal tubes, powered up 24/7. Tube amps have to be turned on and off because output tubes pass a lot of current and will fail relatively quickly if left on 24/7.
The suggestion about reaching out to Glenn Poors is a good one.
