Some theorizing on why the battery can affect the sound....
Disregarding the fact that it does not make obvious sense (which is purely because we do not have the necessary data, or understanding), and looking at the results, what we are seeing is similar to what we would see by cleaning up and improving the power supply on an amplifier. In that situation, it makes perfect sense, and we can easily accept why the sound is different/improved. After all, if you have just spent the time and money on building a much better linear supply, with lots of current reserve, high grade capacitors, low noise rectifiers, and sweated all the details, wiring, low noise transformer, etc., you would expect to hear a difference in the SQ.
So how does this apply to changing one battery for another in a DAP?
Well, the common thing is that we are changing the power supply. One speculation is that the protection circuit may be introducing noise into the 5V output. As there are electronic circuits inline with the output, and able to monitor/control/disconnect the output in case of over current, shorts, etc.. So if there is an electronic circuit inline, and it is introducing noise, impressed onto the 5V output, or Grd, this will cause the DAP digital and analog circuits to waste power, handling the noise, or actually decoding/amplifying the noise, and filtering it out. Remove the noise, and all of a sudden, the DAP is able to work without wasting power, doing error handling, or wasting CPU cycles. The result is more dynamics, detail, staging, bass, etc.
The above came to mind, as I worked on a problem (many years ago) where telephone central office equipment was periodically locking up. 5 pairs of redundant processors would lock up, and after pulling everything apart, inspecting and finding nothing, it would start working normally again, until the next time it happened. We had already changed the hardware more than once, so we had to look for another cause. It turned out to be a noisy switching power supply. The electrolytic filter capacitors had degraded, and was introducing a higher than normal ripple. We found that about 0.2V of ripple on the 5V output could cause the processors to lock up. And this is not theory, as we reproduced it in the lab.
Since the months we spent on that problem, my eyes were opened to how much impact a noisy supply could have, even though the voltage and current measured fine. I am stretching to compare this battery situation as a similar situation, but it seems to make sense to me, and would explain how a "simple" battery can affect the SQ in such a drastic manner. But of course, a battery is no longer just a chemical cell with two leads connected to it, and nothing else....they are much more complicated than that, nowadays.
So, that's my contribution to trying to put forward a theory that could explain what we are seeing. As with all theories, they are just that until proven right or wrong. But if you don't try to theorize, and trigger other thinking, then the vacuum never gets filled, so I've always believed that speculation is necessary. Denying a fact that you are seeing because it does not make sense, is a dead end. And I can say that from a background of spending decades doing troubleshooting on telephone central office switches.