Hello @Merakidea
Can you please check out with Shipping & Delivery ‑ ShipZip ?
It has option to set a single delivery charge(a flat rate) if order comes from multiple locations.
Easy to setup on the app to create a zone, and a rate with base price and set a single location fee option on the setting page.
App support provides live chat to help you shipping rates.
Also available to test free in test mode.
Just for information: You should have enabled CCS(custom carrier service) on your store to set shipping rates from third party APP. (that comes free with height or advance plan OR annual billing cycle with shopify($79) plan.