I needed to batch compile some files recently, and I had to lookup how to do this with Aquamacs. It’s pretty simple. If Aquamacs is at /Applications/Aquamacs, then this will work:

/Applications/Aquamacs\ Emacs.app/Contents/MacOS/Aquamacs\ Emacs -nw -batch -f batch-byte-compile /path/to/elisp-file.el