The PayflowPro payment processor extension for CiviCRM attempts to include the financial type by including the following item in the data that is sent to the PayflowPro API. Although, I had to modify this a bit so that the financial type is still sent to PayflowPro when a user signs up for an event.
$payflow_query_array = [ ... 'COMMENT1' => urlencode($params['contributionType_accounting_code']), ... ];
Do the other payment processor extensions for Civi do this - send the financial type to the payment processor dashboard? Specifically, the big ones: Stripe, Authorize.NET, and IATS?