The escaping defined in Dockerfile did not seem necessary and it was never finding my map file.
I changed it to this and it worked.
#MS_MAP_PATTERN=^\/etc\/mapserver\/([^\\.][-_A-Za-z0-9\.]+\/{1})([-_A-Za-z0-9\.]+\.map)$
MS_MAP_PATTERN=^/etc/mapserver/([^\.][-_A-Za-z0-9.]+/{1})([-_A-Za-z0-9.]+.map)$