Possible to change the URL for the regular post type without affecting the URL of other custom post types?

You could do this on custom post type registration.

1) Set your default permalink in the WordPress admin to your desired structure e.g.:

/blog/%postname%

2) Add the “slug” and “with_front” parameter to the rewrite-array in the register_post_type function. “slug” must be the name of your post-type.

  $args = array(
    // ...
    'rewrite' => array(
        'slug' => 'your_post_type',
        'with_front' => false
    ),
    // ...
  ); 
  register_post_type('your_post_type',$args);

This should generate the following rewrite-rules:

Default Post: http://example.com/blog/%postname%
Default Page: http://example.com/%postname%
Custom Post Type: http://example.com/your_post_type/%postname%

EDIT

The “slug” parameter is optional. If you don’t set it, the name of your custom post type is used. Check out the function reference about register_post_type:
http://codex.wordpress.org/Function_Reference/register_post_type