Hi There,
I think redoing the entire site is a bit too far, but if that makes you feel comfortable working on the site then you can do that.
However, I see that your PHP memory limit is only 128M (see under Dashboard > WP-Server Info > PHP-Speicherlimit), lets try increasing that first.
To do this, please enter the following lines in your wp-config.php file at the root of your installation:
define( 'WP_MEMORY_LIMIT', '256M' );
define( 'WP_MAX_MEMORY_LIMIT', '512M' );
Add that code just before the line:
/* That's all, stop editing! Happy blogging. */
After you add that check it in Dashboard > WP-Server Info > PHP-Speicherlimit if that did take effect. If not, please contact your hosting on how you can increase the PHP memory limit effectively.
Let us know how it goes,
Cheers!