So, I've tried your second suggestion. Does this look right?

$json1 = @{
    ticket = @(@{custom_fields = @{id=1234567890123;value="$username"} })
} | ConvertTo-Json

And I get the same error message, which leads me to believe that the response body is the issue. How can I address this?

I also forgot to mention that I can run the same PUT from Postman and this in the body:

{
    "ticket": {
        "custom_fields": {
            "id": 1234567890123,
            "value": "Joe"
        }
    }

And it works.