ひとまず名前解決できたという感じ
$ cat > resolv.dnsmasq.conf << 'EOS'
nameserver 8.8.8.8
nameserver 8.8.4.4
EOS
$ cat > dnsmasq.conf << 'EOS'
resolv-file=/etc/resolv.dnsmasq.conf
strict-order
server=/wasabisys.com/198.51.44.5
server=/wasabisys.com/198.51.45.5
server=/wasabisys.com/198.51.44.69
server=/wasabisys.com/198.51.45.69
EOS
$ cat > Dockerfile << 'EOS'
FROM alpine:3.12
RUN set -xe && \
apk --update add dnsmasq && \
rm -rf /var/cache/apk/*
EOS
$ docker build -t wasabimasq .
$ docker run --rm -d --name wasabimasq \
-v $(pwd)/resolv.dnsmasq.conf:/etc/resolv.dnsmasq.conf:ro \
-v $(pwd)/dnsmasq.conf:/etc/dnsmasq.conf:ro \
-p 127.0.0.1:53:53/tcp -p 127.0.0.1:53:53/udp \
wasabimasq /usr/sbin/dnsmasq -d --all-servers
$ dig @127.0.0.1 yahoo.co.jp +short
183.79.135.206
182.22.59.229
$ dig @127.0.0.1 s3.us-east-2.wasabisys.com +short
38.73.225.22
38.73.225.20
/etc/hosts.conf の nameserver を nameserver 127.0.0.1 に書き換えれば名前解決できるようになる。