tl;dr: Talk the client out of their idea. Do the redirect in .htaccess
What your client is suggesting is certainly doable. However, doable != advisable..
That said, it would be most efficient to host both the old site and new site on the same server, and then put the redirect into either the .htaccess or better yet the actual vhost config (if allowed by the new host) on the new server.
If that is not an option (new host wont allow it, or some similar reason) then yes, better to just do the redirect in .htaccess on the old site. Not only to address the possible SEO concerns regarding 301 vs 302, but also because it is more flexible. The only real downside to your approach vs the clients suggestion is the continued billing for the old webhosting. If you can mitigate that by moving the hosting for both to the same place, great, if not, keep both hosting accounts despite the cost.