I'm not sure if there is a way to do this with code or a custom theme. Another member of the community may have more information about that.
Meanwhile, if you are open to using third-party solution, Intuitive Shipping gives you ability to customize your shipping method description. So, while you can't display the handling fee as a separate line item, you can include a custom description that says something like, "Shipping includes $125.00 handling fee." This is displayed with the shipping method, below the delivery time.
If you'd like more information about this or how to set it up, please feel free to arrange a one-on-one demo with one of our team members who will be happy to walk you through the process and answer any questions you might have.
Please note that to use Intuitive Shipping - or any third-party rate calculation app - Shopify requires that you enable their third-party carrier-calculated shipping rates feature. This feature is $20 USD per month. If you change your Shopify billing to annual, they will add the feature at no additional cost (plus they will reduce your billing plan by 10%). If you're on the Advanced plan or higher, this feature is already included and activated by default.