Update custom field by API or hook

Thanks Gui for the hints. I’ve looked into the ‘Update subscriber action’. As far as I can see, I could only update a custom field with a fixed value here:

Couldn’t find anything about inserting something like a shortcode here…
So i went the brutal way, inserting my custom field value directly into the db. That works as intended. If anybody has the same problem, here is part of the code how to do it:

// Prepare subscriber data
        $subscriber_data = [
            'email' => $email,
            'first_name' => $first_name,
            'last_name' => $last_name,
            //'custom_fields' => [
            'cf_2' => $shop_name
            //]
        ];
        if (class_exists(\MailPoet\API\API::class)) {
			$mailpoet_api = \MailPoet\API\API::MP('v1');
			// first see if the user exists
			try {
				$subscriber = $mailpoet_api->getSubscriber( $email );
			} catch ( \Throwable $th ) {
				error_log('getSubscriber: ' . $th->getMessage());
			}
        	// If the subscriber doesn't exist, add them.
			if ( ! $subscriber ) {
				try {
					$result = $mailpoet_api->addSubscriber( $subscriber_data, $list_array );
				} catch (\Throwable $th) {
					error_log('addSubscriber: ' . $th->getMessage());
				}
			} else {
				try {
					$result = $mailpoet_api->subscribeToLists( $subscriber['id'], $list_array );
					// put the shop name directly into the db, as there is no update->subscriberData
					// in the API
					global $wpdb;
					// Define MailPoet table name
					$table_name = $wpdb->prefix . 'mailpoet_subscriber_custom_field';
					$subscriber_id = $subscriber['id'];
					$custom_field_id = 2;
					// Construct and execute the insert query
            		$inserted = $wpdb->insert(
                		$table_name,
                		array(
                    		'subscriber_id' => $subscriber_id,
                    		'custom_field_id' => $custom_field_id,
                    		'value' => $shop_name,
                    		'created_at' => current_time('mysql'),
                    		'updated_at' => current_time('mysql')
                		),
                		array(
                    		'%d', // subscriber_id (int)
                    		'%d', // custom_field_id (int)
                    		'%s', // value (text)
                    		'%s', // created_at (timestamp)
                    		'%s'  // updated_at (timestamp)
                		)
            		);
					error_log('Insert customer shop name in MailPoet: ' . $inserted);
				} catch ( \Throwable $th ) {
					error_log('subscribeToLists: ' . $th->getMessage());
				}
			}
			//error_log('MailPoet addSubscriber/subscribeToLists return info: ' . print_r($result, true));
			// Check for errors
        	if (isset($result['error'])) {
        		error_log('MailPoet subscription error: ' . $result['error']['message']);
        	}
        }