The first problem here that I would address is you have a test case that appears to fail the first time that it is ran and yet passes when it is forced to re-run later. That is the symptom we likely need to address, and not go down the rabbit hole of trying to force ReadyAPI to update/re-generate the test report with a new test suite status. I believe I've seen posts here on the community of people trying it, but I've never delved into that myself.
That being said, do we know why the test case seems to always fail the first run? And pass the second?
Click the Accept as Solution button if my answer has helped, and remember to give kudos where appropriate too!