Our new website has not gone live yet. But based on past experience, we are doing free shipping across the board. UPS and USPS tools all overcharge and it is a nightmare trying to get it right, which we never have. We have a UPS account that is separate from the website that overcharges us too, despite being a loyal account holder for over 15 years. And they do not help with any apps or integrated tools, saying we have to talk to the developers.
One thing we have not done yet, and we may try in the future, is to integrate Pirate Ship onto our websites. They do have a Squarespace option but you would need the Advanced Commerce plan. Pirate Ship is free to use (but you must open an account) and has pretty low rates.
https://ship.pirateship.com/settings/integration
Let us know how it works out if you choose to use it.