You have posed a question I've never heard asked before, because a sampling rate of 11,025Hz would have a theoretical frequency response up to 5,512.5Hz, which is closer to the range of telephone/speech than music.
The 22,050Hz files would have a frequency response up to 11,025Hz, and would be closer to music range.
Although I don't have the BDA-1, most DACs have a similar lower specification, so I will have to try that and see if anything plays. I assume you are using a music player, OS and driver that are bypassing any internal upsampling?
Steve