How can I use a different template file for the blog index’s paginated pages?

You can use this:

$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;

$paged will have the value 1 in you are or the first page or the value of respective pagination page number.

Depending on this you can call get_template_part() to load the desired template for the page.

Let me know if this helps.