With setting the DAC's sample rate I meant setting it on the OS level (e.g. Windows sound settings).
Basically the Amazon app just looks at what bit depth and sample rate are set for the DAC on the OS level, then when you pick a song the app picks the highest quality version that is at that or a lower bit depth and sample rate, and then sends it to the OS, which upsamples if there's a mismatch.
So if you set your DAC (in Windows) to 16 bit, 96 kHz, the app would skip a 24 bit, 192 kHz version of the track and may instead choose a 16 bit, 44.1 kHz version, send that to Windows, Windows resamples it to 24 bit, 96 kHz and sends it to your DAC. If you set your DAC in Windows to 24 bit, 192 kHz it will pick the 24 bit, 192 kHz track and send that to Windows. Windows won't resample or otherwise alter it (with exclusive mode), but may alter parts of it (without exclusive mode) depending on how loud a section is or what other apps are playing audio (to avoid clipping, presumably).
When you then switch to a track that has a 24 bit, 96 kHz version as the highest quality, the app will send that to Windows and Windows will resample it to 192 kHz since that's what your DAC (in Windows) is still set to.