When submitting the form site.com/blog/wp-admin it goes to site.com/wp-admin

You have to define the home url, in the installation folder of site.com/blog:

define( 'WP_HOME',' http://site.com/blog' );
define( 'WP_SITEURL', 'http://site.com/blog' );

I think this can solve your problem.

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