Never mind. I solved it. For anyone interested, my template (in the plugin directory) just does
<?php
acf_form_head();
get_header();
?>
In my plugin file, I made a function like this
function load_acf_form_head() { // do things to get the template };
add_filter('wp_enqueue_scripts', 'load_acf_form_head', 0);
This seems to change the order the template’s called in.