The problem with the register page that's currently displayed (apart from the fact that there's no option to just login), is that it shows all CB registration fields, many of which are hidden by CBSubs in our plans

Have you hidden those fields from registration on the Workflows tab of your plans or simply disabled those fields display on registration?

How do I change the URL of this page?

Please see CBSubs > Setting > Display > Expiration > Membership needed for access redirect URL for existing users:

You can configure the parameter to whatever your like. If the page in question is not the typically access page where a user needs to be subscribed to XYZ to view it then you can not change its URL. The access denied redirect is coded into the CBSubs Bot which is encoded. What it's doing is sending users away from Joomla registration/login to CBs. You can actually change what URL Joomla it self redirects to which might prevent the CBSubs bot from catching it so you can send users to say an article instead. Please see the below.

ARTICLES - HTML
IN: components/com_content/views/article/view.html.php
ON: Line 90
ARTICLES - PDF
IN: components/com_content/views/article/view.pdf.php
ON: Line 57
CONTACTS
IN: components/com_contact/views/contact/view.html.php
ON: Line 71
In the above simply change the $url to whatever you want for example 'index.php?option=com_comprofiler'. Do not include live site in the URL, always start with index.php.

The registration fields have been hidden using the workflows tab.

The fields only hide if the plan is actually selected. If this isn't working then there's probably a fatal JS error on the page; please check your browser error console.