This is a great question and I can see that you have been waiting for a solution that works for your business for a few months now. I can see from your original post that you are aware of some of the limitations of editing the checkout and that you would need to be on a Shopify Plus Plan to access those features.
I did connect with our internal theme support team and they confirmed that the options for editing the checkout and specifically the favicon are not available for merchants on the standard plan. It does require the PLUS plan to access it at this time. I know that wasn't the answer you were looking for but that is the case in this situation.
If you have additional questions about this please don't hesitate to let me know.