If the "something at the other end" sees any URL (i.e. deep page) requested on those other domains redirected to the root of the new site, then you should fix that code to redirect to the right page each time.
Do check that non-canonical requests are redirected once; make sure there is no redirection chain generated for any request. Use the Live HTTP Headers extension for Firefox to check things out.
If the other person is paying for additional hosting merely to place a redirect in that other hosting, change the DNS around to point all of the domains to the server where the site actually resides and cancel the other hosting. In the config of the site where the content actually resides set it to redirect requests for the wrong domains to the right domain, preserving the requested path in that redirect.
I would use this on the actual site:
RewriteCond %{HTTP_HOST} !^(www\.example\.com)?$
RewriteRule (.*) http://www.example.com/$1 [R=301,L]
This ensures that any non-canonical hostname is redirected. It caters for
www.example.com:80
with a port number,
www.example.org
for a different domain,
www.example.com.
with a trailing period (yeah that IS a valid request),
example.com
the bare non-www, or any combination of non-canonical items.