I was able to get my USB headset to work.
First I changed from mame4all-pi on RetroPie 4.5 to lr-mame2000 on RetroPie 4.7.1 because I learned mame4all-pi doesn't have an option to rotate the display and my objective is to play classic vertical arcade games on a vertical (rotated) monitor.
Second I created /etc/modprobe.d/alsa-base.conf (using sudo vi) containing:
options snd_usb_audio index=0
options snd_bcm2835 index=1
options snd slots=snd_usb_audio,snd_bcm2835
(leaving dtparam=audio=on in /boot/config.txt)
This reorders the "cards" so the USB headset is card 0, hdmi is card 1 and PWM is card 2. What I found is dtparam=audio=off still configured the USB headset as card 1 for whatever reason and I didn't get audio. After reordering it worked without any other modifications (although ES gives an error about unable to set volume, which is probably just a minor configuration tweak).