Very often, these facts can be learned by reading the data sheet for the DAC chip and for popular DAC families, the parameters are essentially public information.
Even if you don’t, but the general architecture of the DAC is known (there are only a few), you can make an educated guess about what might help.
DACs usually work in multiple stages.
So for example, the audio may be upsampled 44.1kHz → 88.2kHz with one filter, then 88.2kHz → 176.4kHz with another, 176.4 → DSD64 with a third, DSD64 → DSD256 with a fourth stage. Or something like that.
This is done because it is much cheaper than doing it in one go. This is mainly because after the first stage, very lenient (read: inexpensive) filters can be used.
From a mathematical precision + filter choice point of view, the earlier stages are more important than the later ones. So even if you can’t get all the way up to the final output rate of the DAC in software, upsampling can still bypass the earlier (most important) parts.
I can see how you might think that, but it isn’t consistent with the math. There is ample precision both in DACs and in software based upsamplers to perform their work without pushing arithmetic error anywhere near to an audible place, even with several stages of processing and accumulated error. Especially in software, the stopband rejection of these filters is often well beyond the point where it could–from an arithmetic standpoint–introduce the filter’s noise floor into the signal.