Leave the sample rate in the track at 44100 Hz.
Change the “Project Rate” (bottom left corner of the main screen) to 22000 (or whatever you want). The exported file will be at the same sample rate as the “Project Rate” setting.
Can Audacity export uncompressed 8-bit WAVs? I’m not sure. AFAIK Audacity can only export uncompressed 16-bit and 32-bit WAVs. An uncompressed 16-bit WAV (exported by Audacity), played as 8-bit WAV by the Eletech DP-100 sounds garbled.
Edit: just checked, Audacity_2.0 indeed can export uncompressed 8-bit WAV files.
Have you already tried to type a “Project Rate” of e.g. 11000 Hz (instead of the Audacity 11050 Hz preset) into the text field of the “Project Rate”? Low-CPU players can be extremely picky about such things.