on my hdmi tv, for my PS1, i just ran a cable into video RCA, and ran the audio into an external speaker setup.
with a dreamcast, i ran a rca video to s-video splitter, and again ran audio to the external speakers.
same setup with a nintendo to my son's tv set.
you don't want to upgrade the video...it's 1986, after all.
just adding this, as an hdmi adapter may not give you audio off the snes. good luck.