dynmap through mod_proxy on a external webserver
Solution 1:
You should really check the URI mapping rules for proxy requests.
And overall prefer <Location> to handle proxied content.
Also note the ProxyPassReverse directive in Apache, as well as dynmap configuration for its web resource locations. By default, it is using relative paths, so that all you should've done is
<Location "/map/">
ProxyPass http://192.168.2.31:8123/
</Location>
Just note that in such configuration your map will be dependent on the server being online.