Apache rewrite proxy only homepage

Try something like the following instead, at the top of the .htaccess file, before the WordPress front-controller:

RewriteCond %{ENV:REDIRECT_STATUS} ^$
RewriteRule ^(index\.php)?$ http://some-other-domain.com/ [P]

The RewriteBase and RewriteEngine directives are not required here. The RewriteEngine presumably already appears later in the file as part of the # BEGIN WordPress code block (the order is not important).

Note the slash suffix on the substitution string. This is a required part of the URL (you can't have an empty URL-path). If you omit it here, then something else must "fix" it later. In the case of an external redirect then the browser would "fix" it, but there is no "browser" here.

This matches both an empty URL-path and index.php (in case mod_dir is issuing the subrequest for the directory index before mod_rewrite sends the request through mod_proxy). The condition that checks the REDIRECT_STATUS env var is necessary in this instance to avoid proxying everything, since the WordPress front-controller rewrites everything to index.php. The condition ensures that only direct requests are matched, not rewritten requests.

Aside: A RewriteRule pattern like ^/$ will never match in a .htaccess context. This would only work if the directive was used in a server (or virtualhost) context.