One of the Elastic IPs associated with my running instance is not working, while the other is working.

I think that this is an operating system issue (and not specifically Windows). Your instance will have two private IP addresses with an Elastic IP for each. But it only has one default route. Which means that traffic will generally always try to exit the instance via whichever interface is selected with the default route. This may cause issues where packets will leave a network interface but with a source (private) IP address that is not the same as the network interface - assuming the the operating system will yet you do that (it may not).

You might be able to fix this by disabled source/destination check on your interfaces: https://docs.aws.amazon.com/AWSEC2/latest/UserGuide/using-eni.html#modify-source-dest-check

Why do you need two Elastic IP addresses for a single instance? I get that you're running two environments but I think it's not a good idea to have them both on the same instance. It would be really easy to make a change in UAT that affects production.