Nevermind, I figured it out. The standard lost password action was getting in the way.

inc/location/class-lost-password-form.php

@@ -34,7 +34,9 @@ final class LostPasswordForm extends LoginLocation {
     {
         parent::activate();
         add_action('lostpassword_form', [$this, 'render']);
-        add_action('lostpassword_post', [$this, 'process_form'], 10, 1);
+        if (!isset($_POST['woocommerce-lost-password-nonce'])) {
+            add_action('lostpassword_post', [$this, 'process_form'], 10, 1);
+        }
     }
 
     public function process_form(WP_Error $errors):void {

ogre-captcha.php

@@ -107,7 +107,7 @@ if (!class_exists('Ogre\Captcha')) {
             $this->add_files([
                 'inc/location/woocommerce/class-checkout.php',
                 'inc/location/woocommerce/class-login.php',
-                // 'inc/location/woocommerce/class-lost-password.php',
+                'inc/location/woocommerce/class-lost-password.php',
                 'inc/location/woocommerce/class-register.php',
             ]);
         }