Not that I necessarily think it's best to stay with Postman, but have you looked at newman, which is the CLI runner for postman configs? We had postman as a test suite for something (which is more an API than an app), and I got tired of having to deal with setting up extra steps to test and of exporting the postman config to save in the repo, so I put newman on the test system and just run against the config directly in the test environment and check the output.

I don't necessarily recommend editing the postman config json directly to set up new tests as it's a PITA, but it's generally what I do so I don't need to keep importing and exporting it with Postman.

A tool designed for working with on the shell is likely better than what I'm doing with newman (since the config is not the most accessible), but it also meant I didn't need to rewrite a bunch of existing tests and verify they actually did the same thing.