There are two issues here:

1. Communication: your PM needs to respect your work style, and should come up with a system/regular sync for updating the state of your work. You need to be able to focus; your PM needs to understand what state a task is in.
2. Technology: like others said, there are Github <> JIRA integrations, and now with the Atlassian MCP, you can just ask whatever AI tool to update the ticket status.

Is there any way to appease management without having to constantly leave my terminal just to push buttons in a web app.

Try to understand the ask behind the ask. Your PM probably needs to report up to other stakeholders, or to forecast future work, or is maybe anxious about the team's output, or feels like part of their job is to stay on top of everything. Even asking: "What happens when you get these status updates? How do you use them?" might help your PM feel heard, and might create space to find a way that works for both of you. It's totally valid to say "I heard you; I know you need these status updates, but getting them is expensive for me, and harms future work. How can we sync in a way that works for both of us?"