How can I sort posts ascending by post title for a specific post type, but on a category archive template?

You can use pre_get_posts to hook in and change the query.

The argument passed to pre_get_posts is the query object. It fires for every query, so you need to check some things first. Quick example (not-tested) — change the order on only the video type’s post type archive page.

<?php
add_action('pre_get_posts', 'wpse73991_change_order');
function wpse73991_change_order($q)
{
    if(
        is_admin() || // no admin area mods
        !$q->is_main_query() || // make sure we're modifying the main loop
        !$q->is_post_type_archive('videos') // is this the video archive? (you might need to change this)
    ) return; // bail, we're not where we want to be.

    // change the order
    $q->set('order', 'title');
}