That wouldn't be a unit test - that's more like an end-to-end or integration test.
Have you ever worked anywhere that had perfect test coverage? It just doesn't happen, nor is it possible unless you're building a calculator app or todo list.
Some days, I would seriously rather work at Wendy's.
Narrator: He would not.
Money is not important to me insofar as I have enough to live an average life. I don't need anymore than that.
Or, don't tell them and do it anyway. It'll make your own life 100x easier, unless you enjoy the stress of bug fixing and responding to incidents where you've broken something.
Writing code without tests because it takes too long is like running a car engine without ever bothering to change the oil - it'll run for a bit, and eventually fail catastrophically.
Unit tests already mock network calls so create mocks to return errors instead.