I got the software downloaded and have been able to interact with the API gateway locally, but I’m struggling to get it to work publicly.

MIP
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My first suspicion is Cross-Origin Resource Sharing (CORS) as the cause for your struggle. See the chapter with the heading Cross-Origin Resource Sharing (CORS) here;

https://doc.developer.milestonesys.com/mipvmsapi/api/config-rest/v1/#section/Introduction/Configuration